Statistical modelling of gambling probabilities

Transcription

1 Ttle Statstcal modellng of gamblng pobabltes Autho(s) Lo, Su-yan, Vcto.; 老 瑞 欣 Ctaton Issued Date 992 URL Rghts The autho etans all popetay ghts, (such as patent ghts) and the ght to use n futue woks.

2 STATISTICAL MODELLING OF GAMBLING PROBABILITIES Vcto Su Yan Lo A thess pesented towads the degee of Docto of Phlosophy Unvesty of Hong Kong June 992

3 Ths thess s submtted to the Unvesty of Hong Kong fo the awad of the degee of Docto of Phlosophy. The whole wok has been undetaken afte egstaton fo the degee and has not been pevously ncluded n a thess dssetaton o epot submtted to t h s o any o t h e n s t t u t o n f o a degee, dploma o othe qualfcaton- Vcto S.Y. Lo

4 ACKNOWLEDGEMENTS The subject supevso, D. matte J. of ths thess Bacon-Shone, Lectue was of suggested the by my Depatment of Statstcs.Decto of Socal Scences Reseach Cente, Faculty Dean of Socal Scences, Unvesty of Hong Kong and my nfomal advso, D. K. Busche Lectue o f t h e School o f Economcs, Unvesty o f Hong Kong. I must t a k e t h s oppotunty t o e x p e s s my deep g a t t u d e t o both o f t h e m f o t h e concen, suppot, contnuous gudance and enlghtenment thoughout t h e e n t e peod o f study. The good deas, stmulaton, p a t e n t coectons and encouagement made by m y supevso w e e ndspensable t o t h s t h e s s. I am a l s o g a t e f u l t o D. W.K. L and D. Y.C. Kuk f o t h e advce. Moeve, I would l k e t o thank D. K. Busche, P o f e s s o s M.M. Al and R. Quandt f o t h e data, and P o f e s s o Junj Shba f o h s h e l p n acqung t h e Japanese data.

5 Abstact o f thess enttled "STATISTICAL MODELLING OF GAMBLING PROBABILITIES" submtted by Vcto Su Yan Lo fo the degee of Docto of Phlosophy at the Unvesty of Hong Kong n June 992 Economsts and psychologsts have long shown Inteest n acetack bettng as a souce fo nvestgatng atttudes to sk and effcency of makets. Most of the pevous studes have concluded that bettos undebet favoute hoses and ovebet longshots. Hence, the bettos may be sk-loves. A dffeent phenomenon n Hong Kong acetack bettng was also epoted. Howeve, to date the statstcal methods have been faly udmentay and thus, thee s oom fo mpoved analyss. In ths thess, we popose and apply new statstcal methodology to study the bettng maket usng a lage amount of data obtaned n dffeent acetacks. A smple model s developed fo ths pupose. The conclusons fo the wn bet n dffeent acetacks ae consstent wth the pevous studes. Apat fom analysng the smple wn bet, we extend ou model to nclude moe complcated bets such as the exacta.

6 Usng wn bet factons alone, models fo pedctng odeng pobabltes based on dffeent dstbutons (exponental, nomal assumptons and of gamma) unlng ae tme compaed empcally and theoetcally. Patcula vaance and coelaton stuctues of unnng tmes ae also consdeed. As most of these models ae dffcult to apply n pactce, a smplfcaton of these sophstcated models s poposed. The smplfed models ae then appled to a bettng stategy (D.Z s system} n dffeent a c e t a c k s. Results ndcate t h a t t h e e s a n mpovement ove t h e ognal s t a t e g y.

7 To my Dad who neve loses because he neve bets and my gandmom who s a wnne because she ases me I

8 Ths thess s submtted to the Unvesty of Hong Kong fo the awad of the degee of Docto of Phlosophy, The whole wok has been undetaken afte egstaton fo the degee and has not been pevously ncluded n a thess, dssetaton o epot submtted to ths o any othe nsttuton fo a degee, dploma o qualfcaton. Vcto S.Y. Lo othe

9 CONTENTS Page Chapte Intoducton LI Backgound ].2 Racetack bettng makets ;.3 Reseach aeas - L4 Data souces Chapte 2 Lteatue evew 2. Intoducton Rsk behavou and maket effcency Estmaton of outcome pobabltes Optmal bettng Chapte 3 Modellng the wnnng pobabltes 3. Intoducton A s method of analysng wn bet data Classes of multnomal logt models 33 3A 3.5 Empcal analyses n the U.S. H-K-, Japan and Shangha Compason w t h t h e othe models Relatonshp between t h e b e t a and pool s z e s Monng l n e odds and e a l e odds 68

10 3.9 Conclusons Chapte 4 72 Analyses of moe complcated bets 4. Intoducton Descpton of some poposed models Extended logt model fo moe complcated bets Empcal analyss fo exacta bet Empcal analyss fo tfecta bet Empcal analyss fo qunella bet Empcal analyss fo double bet and double qunella Conclusons 04 Chapte 5 Detaled compason between the Havlle and Heney models 5. Intoducton The motvaton Condtonal logstc analyss fo the Havlle and Heney models Theoetcal nvestgaton of the Havlle and Heney models Concluson 23 Chapte 6 Extensons of the Heney model 6. Intoducton The selected model 25

11 6-3 Appoxmaton fomulas 3 6A 33 Chapte 7 A ecommended smple model 7. Intoducton Smlaty between the two models Some smulaton esults Empcal analyss usng fxed lambda and tan Matchng between the Heney model and exacta bet f Facton Othe appoxmatons fo the Heney model Compason of pobablty estmatons usng a closeness measue Concluson 64 Chapte 8 The Sten model n Japan 8. Intoducton Falue of the Heney model n Japan Fttng the Sten model A smple appoxmaton of the Sten model A theoetcal esult fo the compason between the Havlle and Sten models 8.6 Concluson Chapte 9 Study of bettng stategy 9. Intoducton 93

12 9.2 Descpton of D.Z's system Applcatons of D.Z's system Specal consdeaton n Japan Recommendatons 24 Chapte 0 Oveall conclusons 0. Bettng behavou Complcated bets analyses Models fo pedctng odeng pobabltes Bettng stategy 28 Appendx A Estmated paametes fo A-class and L-class models 220 Appendx B Gaussan-Hemte ntegatons fo complcated pobabltes 226 Appendx C Devaton of appoxmaton fo pobabltes unde extended model 228 Appendx D Devatons of second-ode appoxmatons fo the Heney model Bblogaphy

13 CHAPTER ONE INTRODUCTION. Backgound Gamblng behavou s one topc n decson unde sk. It s elated to at least two Economsts outcomes ae (e.g. Psychologsts nvestgate Kahneman academc felds - Economcs and Psychology. nteested n fndng possble Gethe S c Plott & Mathematcal the behavoual S c Tvesky (979 (979) Psychologsts, pocess (e.g. easons fo the Wetzman (965)). and on the Luce othe (965 & hand, 977), S c 983) and Tvesky & Kahneman 974 Se 98)). Nowadays, Statstcans can wok w t h Psychologsts and Economsts n studyng t h e human decson pocess and explanng vaous phenomena wth s ta t s t c s. Fo nstance, subjectve pobablty and objectve pobablty estmaton c a n be teated as s t a t s t c a l poblems. Racetack bettng has long b e e n ecognsed b y Economsts and Psychologsts as a souce of nfomaton f o nvestgatng atttudes t o sk and e f f c e n c y of makets. Racetacks and s e c u t e s makets have many chaactestcs n common. One k e y d f f e e n c e s t h e complexty; t h e acetack s e a l l y a sequence of makets t h a t ae e l a t v e l y smple and shot-lved.

14 To bettos, of couse, t s moe nteestng to pedct the poft. Bettos usng a bettng system attempt to fnd the maket neffcences and bet on such hoses when they have postve expected poft. To date statstcal methods have been faly udmentay. We have poposed mpoved methods and used these to analyse hose-acng data fom dffeent countes ncludng the U.S., Hong Kong, and Japan n bette ways..2 Racetack Bettng Makets Befoe ntoducng ou study, t s helpful to befly descbe the acetack bettng makets. Usually, fo each ace, bettos have a choce of bettng types. Fo example, wn bet means a betto wll have a postve etun f the hose he bets on wns place b e t means a b e t t o w l l have a p o s t v e e t u n f t h e h o s e h e chooses f n s h e s f s t o second; b e t t n g on t w o h o s e s t o e x a c t a p a y s o n l y w h e n t h e t w o h o s e s f n s h f s t and second n e x a c t ode, e t c. Cwe adopt t h e temnology o f t h e U.S. acetacks). Sepaate pools o f money ae kept f o each bet type. The numbe o f h o s e s n a a c e u s u a l l y a n g e s f o m 6 t o 2 n t h e U.S., 3 t o 4 n Hong Kong and 6 t o 8 n Japan. The smplest b e t t y p e s t h e w n bet. Unde t h e pamutuel b e t t n g system (o t h e totalze system), t h e odds (.e. pofts pe dolla b e t t o a

15 successful betto) o f the w n bet a e detemned f o m the bets made by the publc as follows : In a patcula ace, let Unde the pamutuel system, 0 = (-t) W - X X ^ =odds on hose. s calculated as follows =,2,..., whee t = the tack take, n = numbe o f hoses n t h e a c e, = t h e w n b e t amount on ho se, and W = t o t a l w n b e t amount on a l l h o s e s = S X The popoton o f money b e t on a h o s ex / W s called t h e w n b e t f p a c t o n. Some e s e a c h e s have suggested t h a t t h e w n b e t f a c t o n c a n b e used as a n e s t m a t e o f t h e t u e wnnng pobablty o f a h o s e. One eason s t h a t unde t h e assumpton of p o f t m a x m z a t o n o f bettos, t h e e x p e c t e d e t u n s f o a l l t h e hoses n t h e a c e a e t h e same and t c a n b e e a s l y s h o w n t h a t t h e bet f a c t o n s equal t o t h e t u e w n n n g pobablty. The t a c k take, t s t h e commsson pad t o t h e Jockey club. Some p a t o f t h e take may b e pad t o t h e Govenment n t h e f om o f t a x. T h e s e tack takes a e t y p c a l l y n t h e a n g e o f 5 t o 25 %. It s k n o w n t h a t n t h e U.S. and Hong Kong, t h e t a c k t a k e s f o t h e w n b e t a e about but n Japan, t h e t a c k t a k e f o t h e w n b e t Is appoxmately 26 %. Moe complcated b e t s (usually c a l l e d e x o t c

16 bets) wll have a hghe tack take. In addton, thee s a cost called Beakage n the U.S. Ths addtonal cost s due to the fact that all etuns pe dolla bet (.e. odds plus one) ae ounded down to the neaest fve o ten cents n the U.S. and thus t s not substantal. All changng wn odds ae shown clealy on a bg sceen befoe a ace stats. In Hong Kong, qunella odds ae also vsble. Howeve, fo moe complcated bets (e.g. tfecta - bettng on the hoses fnshng fst, second and thd n coect ode), bettos cannot obseve the odds. Anothe bettng system s the fxed-odds system. It Is countes such as Btan, Austala, Italy and used n Fance. The bookmakes set fxed odds that the bettos can place a bet. But the odds offeed change ove tme. Ths allows the bookmakes to balance the books so as to ensue themselves of a poft egadless of whch hose wns. In ths case, we have no explct estmates of the wnnng pobabltes. Howeve, ths system wll not be elated to the analyses n ths thess snce the bettng systems of ou data sets ae all pamutuel,.3 Reseach Aeas Ou study can be dvded nto thee pats. Fo each pat, we have absobed some knowledge fom the pevously publshed wok.

17 .3. Wnnng pobablty modellng Al (977) and Asch, Malkel 8«Quandt (982) concluded t h a t t h e b e t t o s behave a s sk-loves based on t h e smple analyses. Ths s because a favoute-longshot bas w a s obseved n t h e w n bet. Absence o f t h e bas w a s also epoted n Busche and Hall (988) usng a d f f e e n t data s e t. We have analysed Al^s data s e t by f t t n g a c l a s s o f multnomal logt model and f n a l l y a smple model s suggested. Ths smple model s f t t e d n othe data sets. Ou e s u l t s a e qute consstent w t h t h e e s u l t s o f pevous studes. Ths p a t s epoted n Chapte t h e e..3.2 A n a l y s e s of complcated bets and estmaton of complcated pobabltes Followng t h e smple model obtaned n t h e pevous pat o f ou study, an extenson of t h a t model allows u s t o analyse moe complcated b e t s l k e e x a c t a, t f e c t a and qunella. Estmaton o f moe complcated pobabltes can be based on models poposed by Havlle (973), Heney (98) and Ste n (990). These models a e t e d on d f f e e n t data s e t s. Results n some acetacks suggest t h a t t h e Havlle model s t h e w o s t one but t s most commonly used because of ts smplcty. Moe theo et ca l e s u l t s a e obtaned empcal esults and some f o t h e compasons between d f f e e n t models. These a e pesented In detal n Chapte fou and f v e. Extensons o f t h e Heney model a e dscussed n Chapte s x.

18 Moeove, a smple model s poposed to appoxmate the Heney model n Chapte seven. The Sten model s found to f t well n Japan and ths esult togethe wth ts appoxmaton s dscussed n Chapte eght,.3.3 Study of a bettng system Hausch, Zemba Sc Rubnsten (98) and Hausch & Zemba (985) suggest a system known as D.Z's system. They clam that the system s moe opeatonal and easonable than some tadtonal bettng stateges (e.g. Epsten (977) Isaacs Wlls (964)) whch usually (953) Rosne (975) and assume p e f e c t knowledge of wnnng pobabltes. Fo t h e estmaton o f complcated pobabltes, t h e y u s e t h e model poposed b y Havlle (973). The system optmses t h e b e t amounts f o place and show b e t s usng a nonlnea pogammng model. We have appled D.Z's system n ou data s e t s and ecommend u s n g t h e model poposed n Chapte s e v e n and eght f o estmatng t h e complcated pobabltes. Ths p a t o f t h e study s epoted n Chapte n n e. P a t one (Chapte thee) manly deals w t h a n economc poblem o f s k p e f e e n c e b y f t t n g s t a t s t c a l models w h l e pat t w o (Chaptes f o u, f v e, s x,seven and eght) s concened w t h t h e s t a t s t c a l dstbutons o f unnng t m e s o f hoses a s w e l l a s t h e ac cua ces of d f f e e n t b e t t y p e s. The l a s t p a t (Chapte nne) s c e t a n l y a n nteestng aea f o t h e geneal publc (bettos) a s w e l l a s f o academcs such a s s ta ts tc a n s, opeatonal eseaches

19 and economsts..4 Data souces We have obtaned some hose-acng data sets fom vaous acetacks n the U.S., H.K. and Japan. The detals ae as follows : Race t a c k () A l ' s d a t a data avalable yea no.of aces : (U.S.) Saatoga wn R o o sev e I t wn Yonkes wn ( ) Quand t ' s data Meadow ands : (U.S.) w n p a c e, s h o w, 84 70S ex a e t a, t f e c t a, double bet A t l a n t c Cty wn ( ) Hong Kong d a t a Shat n & wn,place, Happy V a l l e y qunella, 8-90 >4000 d ou b l e b e t s, double qunella, t fe c t a (v) J a p a n e s e data (v) Dffeent w n,place, a c e t a c k s a u n e l l a Old Chnese a c e t a c k data Sh a n gh a :

20 Moeove, the above acetacks have dffeent specal popetes. Fo examples, the pool szes In H.K. ae extemely lage elatve to all the othe tacks; the wn bet tack take n Japan s appoxmately 26% but t s only about 7-87«e l s e w h e e.

21 CHAPTER TWO L I T E R A T U R E REVIEW 2. Intoducton In ths chapte, we wll dscuss some pevous elevant studes. Pevous studes about hose-acng have been publshed n jounals elated to a wde ange of subjects - Statstcs, Pobablty, Mathematcs, Management Scence, Economcs, Busness, Fnance and Psychology. It s mpotant to pont out that the bettng systems beng used n Noth Ameca and Hong Kong ae pamutuel systems. Howeve, n Btan and Austala, both pamutuel system and fxed-odds systems ae beng used. (Fo defntons of these two systems, see Chapte one.) In Japan to ou knowledge, only p a m u t u e l s y s t e m s u s e d and n o e l e v a n t l t e a t u e s f o u n d Rsk behavou and maket e f f c e n c y 2.2. U n t e d S t a t e s and Hong Kong : B e t s a t t h e U.S. and H.K a c e t a c k s a e p e f o m e d unde a pamutuel s y s t e m t h a t c a l c u l a t e s and automatcally updates b e t t n g odds b a s e d o n a l l w a g e s. Gffth (949), eseache n the f e l d a of psychologst, s pobably the fst h o s e - b e t t n g behavou. He c l a s s f e s

22 hs data by odds and compaes the numbe of wnnes n an odds-goup (Wn ) multpled by tack-odds一phs one (+0 ) w t h t h e h h numbe o f h o s e s n t h a t goup. Ths s equvalent t o compang w n n n g f e q u e n c e s and subjectve wnnng pobabltes. Ths s b e c a u s e f t h e tack take s gnoed and t h e subjectve wnnng p o b a b l t e s a e good es t m a t e s o f t u e wnnng pobabltes, t h e n Wn (+0 ) ^ Numbe o f hoses n t h e goup h. He concludes t h a t t h e h h fnal odds chances o f ae, wnnng. t h e ch a n c es o f Anothe hose-acng on t h e aveage, accuate e f l e c t o n s o f t h e But t h e e s a systematc undeevaluaton o f f a v o t e s and oveevaluaton o f t h o s e o f longshots. psychologst, data McGlothln (956) analyses s e t. He also c l a s s f e s t h e d a t a b y t h e n f n d s out t h e e x p e c t e d e t u n ( wt h tack t a k e and coected). a odds lage and Beakage The e x p e c t e d e t u n s l o w when odds a e hghe (Ths e s u l t s m o e s g n f c a n t n t h e l a s t a c e f o e a c h a c e day). Hence, h e concludes t h a t t h e goup o f b e t t o s s sk-takng, n geneal. Two s t a t s t c a n s, Hoel elablty o f subjectve Se Falln (974) demonstate t h e e s t m a t e s usng h o s e - a c n g data. They s e p a a t e t h e a c e s z e s f s t and t h e n u s e Ch-squae t e s t s t o test t h e pecson espect good to the but subjectve o f s ubj e c t v e pobabltes o f wnnng w t h obseved f e q u e n c e s o f wnnng. The e s u l t s a e t h e e s a s l g h t t e n d e n c y t o ovepedct a t l o w e pobabltes. The e s u l t s smla t o t h e conclusons 0

23 of othe eseaches. Al (977) uses a vey lage data hose-acng. He uses the odds of hoses to set to analyse fnd the bet factons (whch ae not avalable n hs data set). He agues that hoses should be classfed by favoute poston (favoutsm) because only one hose fo each goup exsts n each ace. He woks out the aveage bet facton and obseved fequency of wnnng fa each favoute-goup. He assumes that each hose caes a tue wnnng pobablty tt. and defnes the wn bet facton fo each hose as the subjectve pobablty Smple Z-tests ( ae pobabltes a e hghe & then 之 0 and 2 7^= 2 used to show that ) subjectve (lowe) than obseved fequences f o longshots (favoutes). We call t h s knd o f analyss Al's table. Detals o f ths method w l l b e dscussed n Chapte two. A theoetcal explanaton f o t h s e s u l t s povded n h s pape but f o t h e case of t w o hoses only (based on some assumptons). In addton, t h e u t l t y f u n c t o n appoach and delta-measue a e used t o show t h a t bettos ae "sk-loves". Lastly, Al concludes t h a t t h e sk behavou s elated t o captal held by studyng t h e d f f e e n c e o f behavou between bettng on t h e l a s t a c e n a meetng day and othe aces. Anothe eseache, Snyde e f f c e n c y o f t h e hose-acng (978a,b) maket. tested An e f f c e n t the maket

24 means a pefectly compettve maket whee pces eflect all avalable nfomaton. Snyde tes the weak fom test (whethe knowledge about the used to ean (whethe any subjectve an above odds assgned by bettos can be aveage etun) and stong fom test specal goup of people can outpefom the othes n eanng poft). Fo the weak fom test, Snyde analyses sx dffeent sets of data and shows ate of etun gaphcally and sometmes statstcally that the deceases when the odds ncease. Ths fndng s consstent wth Al (977) s. In h s stong f o m t e s t, Snyde shows t h a t nealy all of the e x p e t s ' odds dveged moe f o m an unbased pedcton than dd the geneal bettng publc's pamutuel odds (o subjectve pobabltes of wnnng). Fnally, between the h e concludes t h a t w h l e s g n f c a n t subjectve dffeences and empcal pobabltes of wnnng f o patcula odds-goups o f hoses e x s t, t h e s e d f f e e n c e s ae n o t so lage a s t o exceed t h e pce o f bettng - t h e tack take, Al (979) t e s t s t h e equalty o f etuns between t w o smple b e t s (palay and double bet) o f unknown but dentcal wnnng pobabltes. He concludes t h a t s g n f c a n t l y d f f e e n t and thus they the ae s a n mplcaton of an e f f c e n t maket. 2 t w o etuns a e not "equally pced" whch

25 Fglewsk (979) uses a multnomal logt model to elate the obseved fequency of wnnng to the handcappes' nfomaton and fnal odds. He concludes that do contan consdeable nfomaton dscount almost t. do not all of dscount accuately the the handcappes' but that the Howeve, off-tack subjectve handcappe advce tack bettng odds systems nfomaton as as do on-tack bettos. Regadng Snyde's analyss, Vannebo (980) comments that the skewness of ate of etun s also mpotant n addton to mean and standad devaton of ate of etun. He suggests that the based expected etun s a esult nheent n atonal behavou towads sk and wll be ncued even when the wageng maket s "effcent". In fact, the mpotance of hghe moments was dscussed by some pevous eseaches (e.g. S.Tsang (972)), Losey and Talbott (980) e-analyse Snyde^s data and conclude that bettos tustng the handcappe ae not only unable to get above aveage etun but they may also get etuns lowe than the aveage. The esult s smla to and stonge than Fglewsk's. Asch, Malkel & Quandt (982) analyse the data set smla way to Al. The esults ae also smla bettos ovebet on longshots and undebet behavou s elated favotes to captal held. Moeove, estmated ates of etun 3 to they n a Al's,.e. and sk show that ae lowe f odds ae hghe. Ths esult

26 s smla to Snyde(978). Futhe, they show that the fnal pamutuel odds ae bette than monng-lne odds n estmatng the pobablty of wnnng. Asch, Malkel & Quandt (984,86) use the multnomal logt model to f t the wnnng fequency on vaous handcappng Infomaton. They conclude that poftable stateges fo wn bets cannot be devsed on the bass of maket data. Quandt (986) "Equlbum" theoetcal of fst ceates hs own hose bettng. Then he goes on defnton of to pove some popetes such as the maket cannot be n equlbum f the bet factons and the tue wnnng pobabltes ae the same. Fnally, he concludes that the objectve pobabltes of wnnng ae geate than the subjectve pobabltes of wth the evese beng tue fo wnnng longshots, fo favotes s a natual consequence of equng equlbum to hold n the bettng maket. Howeve, most mpotantly, hs esults ae stongly based on the assumpton of sk-lovng bettos Asch and Quandt (987) analyse exacta and daly double data collected fom Meadowlands n 984. Fo exacta data, they fst ft a lnea egesson of classfed wnnng fequences on subjectve pobabltes (unde the classfcaton by odds). But the assumptons fo lnea egesson ae not satsfed. Then they estmate the objectve pobabltes of success n the j-exacta as follows: 4 ⑴

27 Substtute n the above egesson equaton usng the actual subjectve pobablty of each hose and estmate the coespondng objectve pobablty; (2) Apply the Havlle (973) fomula (dscussed late) to estmate the coespondng objectve pobablty of success h^. Lastly, obtan an estmate of the mplct subjectve pobablty of success fo the j-exacta by substtutng ths * and solve fo s. Also, t h e y compute t h e dect estmate of t h e subjectve pobablty (.e. t h e amounts b e t on each possble e x a c t a combnaton dvded by t h e total exacta pool). Then, they f t a egesson of and * ntecept) ae on and f n d t h a t t h e t w o paametes (slope s g n f c a ntl y d f f e e n t f o m one and z e o * espectvely. Moeove, p a y o f f s deved f o m s^and s " a e compaed and t h e y conclude t h a t t h e p a y o f f s deved f o m a e sgnfcantly hghe. Thus, t h e y "thnk t h e e s nsde nfomaton. All of t h e above a e based on t h e assumpton t h a t Havlle,s f omula s appopate whch s, In f a c t, questonable. The second pat o f t h e pape s analyss o f daly double bets. They compae t h e p a y o f f s t o daly double b e t and palay and conclude s t a t s t c a l l y t h a t t s moe p o f t a b l e t o b e t on t h e daly double. They suggest a possble eason f o t h s f n d n g s t h a t t h e e s nsde nfomaton on t h e daly double bet. Thale and Zemba (988) povde a good summay, dscusson and comments on most pevous Amecan studes n t h s f e l d. They comment t h a t t h e favoute-longshot bas s pobably due t o some 5

28 dffeent easons whch they suggest athe than sk-takng behavou. Fo example, bettos mght oveestmate the chance that the long shots wll wn, bettos may deve utlty smply fom holdng a tcket on a longshot and t s moe fun to pck a long shot to wn than a favoute, etc.. They conclude that modellng bettng behavou s complcated. Bettos' behavou seems to depend on numeous factos such as how they have done n eale aces, and whch bets wll yeld the best stoes afte the fact. Busche & Hall (988 analyse a Hong Kong data s e t usng Al^s t a b l e and conclude t h a t Hong Kong bettos do not undebet f avoutes and ovebet longshots o vce vesa. Ths s vey nteestng and t h e easons f o t h s a e unde nvestgaton. In addton, Busche & Hall (988) dscuss t h e classfcaton poblem. In ode t o t e s t t h e accuacy o f b e t f a c t o n s, t h e pevous e s e a c h e s have t o aange t h e data n such a w a y t h a t t h e hoses w t h n e a c h goup ae "smla" so t h a t t h e wnnng fequency (o estmated t u e pobablty) of each hose-goup can b e computed. As dscussed above, pevous eseaches usually c l a s s f y t h e data n one o f t h e t w o w a y s by b e t f a c t o n s (o odds) o by favoute poston. Howeve, ethe method of c l a s s f y n g data d e f n e s t h e w n odds gven t h e bettng odds. Thee may b e an eo n measung t h e t u e wnnng fequences of epesentatve hoses. Hence, t Is b e t t e t o f n d a method of analyss w h c h s n o t a f f e c t e d b y t h e claussfcato poblem. 6

29 2.2.2 Btan Dowe (976 nvestgated Btsh hose-acng data. s y s t e m u s e d s f x e d - o d d s. We have t w o foecast The types of pces ( o d d s ) p c e (FP, f o e c a s t odds estmated b e f o e s t a t a c e ) and s t a t n g pce (SP, t h e f n a l odds made b y of the bookmakes). D o w e f n d s t h a t t h e coelaton b e t w e e n f o e c a s t p c e and wnnng fequency pce not sgnfcantly less than that between statng and wnnng fequency Then h e concludes t h a t t h e e s nsde wll s nfomaton because f nsde nf o ma t o n no exsts, n o t b e avalable t o t h e publc and only t h e f n a l SP t can eveal t. Howeve, Cafts (985) agues t h a t t h e t e s t u s e d b y Dowe s n o t appopate because a betto can b e t on odds much lage t h a n the fnal statng odds f he gets nsde Inf omaton and t h e e f o e h e can e a n much moe p o f t t h a n t h e aveage. Cafts u s e s h s o w n t e s t s t o conclude t h a t t h e Btsh f x e d - o d d s enables poftable abtage at pces dffeent f o m system SP I.e. p o f t a b l e nsde bettng not avalable t o SP b e t t o s e x s t s. Howeve, t h e s e t w o s t u d es on nsde n f o m a t o n and maket e f f c e n c y only apply t o t h e Btsh f x e d - o d d s s y s t e m but not t o pamutuel system n t h e U.S. o Hong Kong, Heney (985) f s t f n d s t h a t t h e a t e o f e t u n Is lowe If 7

30 the SP odds a e hghe (Ths e s u l t s, agan, s m l a to Snyde (978)). He explans t h s b y makng a n nteestng hypothess that Q=fq, w h e e Q=subjectve estmate o f losng pobablty by the betto, q s the t u e chance o f losng and - f s the f a c t o n dscounted of hs loss. He shows gaphcally and statstcally that but less than L Howeve, he only tes out of data (whch ae not lage). Q =f q h h f s vey close to ths model on two sets Also, f : f o a l l hoses h n a ace then n- = S Q = f 2 q = f (n-) h h h h whee n = no. of hoses n the ace, whch mples that f s constaned to be one Austala : Lke Btan, both bettng system the fxed-odds system and the pamutuel ae used n Austala. Tuckwell (983) analyses odds data fom Sydney and Melboune aces n 974. He obtans a sgnfcant pecentage loss of quadatc can but between sample bettos and the statng pce odds. He dscoves that f gambles bet on hoses begnnng elatonshp "fm" whch sgnfcantly obtan pofts. As the amount ae to be favotes, then they of money of gambles s lage, the oveall magn favoute hoses wll not shot-odds at the taken on ths goup of bookmakes fo the be below the aveage" magn. Howeve, on a 8

31 p o gounds, he expects that the bookmakes should have moe expetse n assessng wnnng chances. Because o f the above fndng, Tuckwell suggests that the only possble explanaton s the exstence of nsde nfomaton may ethe channels of communcaton "nsde" nfomaton. Ths be the consequence of mpefect and/o delbeate ntefeence n ace-outcomes, whch can occu n vaous ways. Fnally, he tests the weak fom and sem-stong fom effcency and fnds that both ae neffcent. Bd, McCae and Beggs (987) analyse data of fom the Melboune acetack n a smla way subjectve vesus objectve pobabltes favotsm). They also analyse the to Al's f o subjectve bookmakes5 odds each wok (.e. level of pobabltes deved fom dffeent odds shown at dffeent tme befoe the ace stats. The esult s smla to Al's. They ague that t would be unwse f o bookmakes to gnoe the bettos* the bettng behavou when pobabltes mpled by the assessment as adjustng bookmakes expessed the odds. by Hence, the odds soon come t o e f l e c t t h e assessment o f bettos. Next, elatonshp they ty an nteestng analyss : between aveage a t e o f e tu n w t h they t e s t the vaances and s k e w n e s s o f t h e etun by f t t n g a egesson COLS, GLS and then SURE ae ted). The vaance tem has a 9 sgnfcant postve

32 c o e f f c e n t w h c h suggests that bettos demand a hghe expected etun to compensate them fo acceptng a hghe vaances n the etun dstbuton. sgnfcantly The c o e f f c e n t o f the skewness tem negatve whch suggests that bettos ae w l l n g s to accept a lowe expected etun n ode to nvest on a hose that offes a l o w pobablty of hgh e t u n. That means, bettos dslke vaance but they lke skewness The authos ty to gve some explanatons fo ths sk-takng behavou. In summay, ths pape concentates on povdng an explanaton of bettos behavou w t h n t h e famewok o f expected u t l t y theoy. Howeve, t h e y note t h a t expected u t l t y models based on t h e assumptons o f goal dected behavou, atonalty and optmalty, have been e x t e n s v e l y ct c z e d n t h e psychologcal lteatue on judgment and choce behavou. Thus, t h e y menton that e c e n t appoaches on human decson may b e moe appopate othe e.g. pospect t h e o y poposed b y Kahneman and Tvesky (979). 2.3 E s t m a t o n o f o u t c o m e p o b a b l t e s Havlle appoxmaton) (973) that ntoduces makes t an assumpton possble t o obtan Havlle pobabltes assocated w t h any complete outcomes n t e m s o f o n l y t h e wnnng pobabltes. 20

33 Let p [ ] b e t h e p o b a b l t y t h a t h o s e s,.. k 2 k l' 2 f n s h fst secod. kth, e s p e c t v e l y, w h e e k ^ n, t h e no. k of hoses, t h e n Havlle's fomula s : M W. y W p [ y k = vwvy wv.-u whee Then = - pj^], etc. k=2, pobablty of p [, = p [ ]p [ ]/(l-p [ ]). Ths 2 X 2 2 X hoses ^ S c fnshng fst elates and the second, espectvely, to the own wnnng pobabltes. Applyng the assumpton to analyse hs data, that bettos ovebet on longshots and Havlle suggests undebet Futhe, although the obseved fequency of second on favotes. and thd place fnshes ae n easonable accod wth the theoetcal long-un fequences, thee oveestmate the seems to be somethng of a tendency to chances of a second o thd place fnsh fo hoses wth hgh theoetcal pobabltes of such fnshes and to undeestmate the chances of those wth low theoetcal pobabltes. Ths fndng may be explanable. Fnally, he compaes (based on hs fomula) wth aveage expected payoff pe dolla the aveage actual payoff pe dolla. The esults show that some of them have sgnfcant dffeences and thus hs assumpton may 2 not hold.

34 Plackett (974) ases the f o l l o w n g "geneal" queston : I f p s the p o b a b l t y that appeas f s t, s the pobablty appeas f s t and j appeas second and p jk ae smlaly defned, how many "levels" of these pobabltes ae equed know all pobabltes hose-acng stuaton. all pemutatons. Cetanly, one case but hs dscusson Is fo the geneal He suggests maxmum smple s wth to to use a logstc model to fnd out the "level" equed. He then apples the method to two vey examples. Howeve, the hoses ae usually dffeent n dffeent aces and thus the estmaton method mentoned n ths pape s not useful n ou hose-acng data sets. Although Havlle egadng the Zemba tue queston & Rubnsten and apply (whch wll the pape, of Havlle's but they (973) not condtonal have a fm concluson ndependence, Hausch, (98) nevetheless assume ths elaton s t to the well-known optmal stategy of bettng be dscussed they n Chapte eght). also poduce pape, slghtly of does.e. bettos In the fst pat of esults smla to the fst pat undebet favotes on wn bet ovebet favotes on place and show bet, etc. Havlle's f o m u l a s c t c z e d b y Heney (98). He develops anothe theoy and woks out anothe appoxmaton fomula f o t h e pobabltes t h a t t h e j t h hose f n s h e s kth, e t c. f a n Independent nomal dstbuton s assumed f o t h e mnng t m e o f each hose. Hs method f s t detemnes estmates o f t h e mean unng tmes f o m 22

35 the the w n n n g pobabltes ( f known o estmable). Then, based on these estmated mean unnng tmes, we can obtan the appoxmate pobabltes f o any complcated bet. Hs esult s qute nteestng but he has not povded any empcal suppot w t h hose-acng data. We w l l dscuss Heney's model n detal n Chapte fou. McCullogh 8 c Z j l (986) gve a "dect" test fo the Havlle fomula by usng the Austalan data. The data set has a popety that the payoff pe dolla to a show bet s ndependent of whch othe hoses n the ace also show and thus the bet facton can be used as a dect estmate of the pobablty of showng. Ths estmate s compaed to the estmate obtaned by the Havlle fomula- That s why they call t a "dect" test. Howeve, they make an mpotant assumpton that ths bet facton s a hghly accuate estmate fo the pobablty of showng. Futhe, they use an eo ndex whch s the pecentage of eo fo the Havlle fomula fo the pupose of compason. Smple lnea egesson s then ftted to these two estmates. The esults ndcate that the Havlle fomula consstently undeestmates the tue pobabltes of showng fa low values and oveestmates the tue pobabltes undeestmates the of showng tue fo hgh pobabltes. values* The Oveall, amount of t the undeestmaton s found to vay consdeably but on aveage to be small. Fnally, dummy vaable egessons ae used to check whethe

36 dstance, feld sze, tack condton, class stakes and dates a f f e c t t h e e o n d e x. They f n d t h a t t h e undeestmaton tends t o b e smalle f o mddle t o long dstance aces fo lage f e l d s z e s and f o t h e w n t e months. Asch and Quandt (987) also u s e t h e Havlle f o m u l a f o analysng e x a c t a data (see s e c t o n 2.2. above f o detals). S t e n (988, 990 does n o t consde Heney's Nomal unnng t m e s model. Instead, a s Havlle's model s equvalent t o assumng ndependent Exponental unnng t m e s, h e suggests t o e x t e n d t h e Exponental dstbuton t o a Gamma dstbuton f o t h e unnng t m e s w t h a f x e d shape paamete () and d f f e e n t s c a l e paametes (¾.^) v a y n g accodng t o t h e mean unnng t m e s o f t h e hoses. Howeve, h s f o m u l a f o estmatng complcated pobabltes h a s no closed f o m, s d f f c u l t t o c a lc ula t e, and n o smple appoxmaton s gven. Based o n a v e y small numbe o f a c e s, h e empcally shows t h a t t h e f o m u l a w t h = 2 s b e t t e t h a n t h a t w t h = l w h c h educes t o Havlle's model. In f a c t, a s m l a d e a h a s aleady been dscussed n Heney (983) b u t Heney does n o t d e c t l y consde t h e h o s e - a c n g poblem and smlaly no s m p l f e d f o m u l a h a s b e e n gven. 24

37 2.4 Optmal bettng Hausch, Zemba and Rubnsten (98) fst show vesus subjectve pobabltes of wnnng, placng (smla to Havlle). "neffcences" ae They suggest that fo not and wn suffcently lage to the actual showng bets, make these postve pofts but etuns fo place and show bets may be possble. Based on the Havlle fomula, they fnd the fomulas of expected etuns fo place and show. The bettng model s : f the estmated expected etun s geate than a cetan value, fnd the optmum expected amount log of place and show bets by maxmzng the etun - the gowth ate of captal o the Kelly cteon (see, e.g. Beman (960)). Howeve, ths s so complcated that the bettos ae unable to fnd the soluton. Thus, they smplfy the pocedue by fttng egessons f o expected etun and optmal bets, s good espectvely. Fnally, they vefy the method by usng actual data and also show statstcally that the success s not andom. Hausch and Zemba (985) extend the 98 Ideas to complcated stuatons. As Zemba s one of the nventos of moe ths pocedue, the stategy s called D. Z's system. Followng the above eseaches, Bolton and Chapman (986} buld up anothe wageng system whch contans two components : a model of the hose ace pocess and a wageng stategy. A model of

38 the hose ace pocess pedcts the pobablty of each hose wnnng f o each ace. The model used s a Multnomal Logt model wth nput vaables weght, ncludng measues of a hose s -qualty, p o s t poston, jockey's c h a a c t e s t c s, e t c.. Usng the "pedcted" w n n n g pobabltes, t h e y e v a l u a t e d f f e e n t bettng stateges (e.g. stateges b y whch the bettos may ean pofts. suggest the to Rosne (975)) and t h e n suggest some bette Fnally, they l a t e e s e a c h c a n combne w a g e n g s t a t e g y smla t o o n e t h e y developed t o g e t h e w t h a t e c h n c a l appoach smla D.U s system. R e c e n t l y, Hausch and Zemba (990) e x t e n d e d t h e D.Z's system t o t h e c a s e o f c o s s - t a c k bettng- That s, t h e s t a t e g y can successfully obtan some pofts by lookng f o coss-tack n e f f c e n c y. Some empcal evdence s g v e n n t h e pape. 26

39 CHAPTER T H R EE MODELLING THE WINNING PROBABILITIES 3. Intoducton In hose-acng, wn bet factons have long been consdeed as a knd of subjectve pobabltes of wnnng snce they eflect the subjectve choces of all the bettos nvolved n the wn bet n a ace. Some people call them consensus pobabltes. whethe the wn bet factons ae good estmates of To study the tue objectve wnnng pobabltes, seveal eseaches n the U.S., the U.K., Austala and Hong Kong have done some empcal studes. Howeve, the statstcal methods wee udmentay. Among these studes, one of the best known s by Al (977). Based on hs method of analyss, he concluded that thee was stong evdence of a favoute-longshot bas,.e. bettos undebet favoutes and ovebet longshots. Howeve, hs method has some weaknesses. In ths chapte w e suggest anothe s t a t s t c a l technque t o a na ly se t h e w n b e t data. A smple l o g t model s poposed a f t e gong though a modellng pocess. Ths model, b y Cox's t e s t, s p e fe e d t o some pevous models. Empcal e s u l t s a e obtaned f o s e v e a l a c e t a c ks n t h e U.S., Hong Kong, Japan and Chna. 27

40 3.2 A l l ' s method of analysng wn bet data Fo each hose ace, Al (977) classfed the hoses by favoute poston (favoutsm). Let P be the bet facton of hose In ace, n so that P whee n j=l = total = 2 0 V and and u Y! P = j numbe of hoses n ace. Classfyng by favoutsm smply means odeng the bet factons as follows : p ⑴ that > P (2) > > p Is, hose (n ) (3.) V () s the most favoute hose and (n ) s the longshot. If the odds on hoses ae avalable but the bet factons ae unknowns (ths was tue n AlFs data set), Al suggested the followng : Let 0 = odds on hose n ace Recall that unde the pamutuel system, Ot s calculated as follows ( VSc ⑷ whee t = the tack take. 28 (3.2)

41 X = the w n bet amount on hose n ace, and total wn bet amount n ace E X and thus, P Fom (3.2), we have (3.3) V & Howeve, n the U.S., Is usually ounded downwads to cents and ths oundng effect 0 Is called Beakage. Due to ths beakage effect, (3.3) does not satsfy the unt sum constant : S P =. To satsfy ths constant, Al followng fomula to estmate the P suggested usng the : /( + 0 I I (3.4) V Sc E /( + 0 Aveage b e t f a c t o n f o t h f a v o u t e ⑴ CD (3,5) whee total numbe of aces n the data set. We also defne n hose n ace and u as the objectve wnnng pobablty fo (I ) as the objectve pobablty fo th favoute n ace. Futhe, we defne the followng ndcato vaable :

42 广 彳 f hose wns the ace 0 othewse Obvously, thee s only one Also, Y () s the ndcato vaable assocated wth P () The obseved wnnng f e q u e n c y f o t h e t h f a v o u t e s : Y = (n 去 ^ =l V (), (3 6) - Al expected the obseved fequency should be a bette estmato of the tue wnnng pobablty f m s suffcently lage because we do not know the accuacy of the bet factons. To test whethe the bet factons ae good estmates of the wnnng pobabltes, Al compaed the bet factons and the obseved wnnng fequences usng smple Z-statstcs : V whee s e (Y())=/Y⑴(l-Y⑴ / m (3.7) (3.8) As an llustaton of hs method, Al's data set s analysed by hs method as shown n table 3.L 30

43 Table 3. bet fac wnfeq? ( n , _ , , , (Note : t h e e s u l t above s d f f e e n t f o m t h a t In AH (977), Whee t h e e w e e equal odds b e t w e e n t h e w nnng h o s e a n d some othe h o s e s, h e a l w a y s chose t h e w nnng hose a s a moe f a v o u t e hose, thus basng t h e esults slghtly towads t h e ove be t t ng o f f a v o u t e s e s u l t. We u s e a n a l t e n a t v e method : w e a l t e n a t e b e t w e e n t h e f s t and t h e second h o s e a s t h e moe f a v o u t e w h e n e v e thee s a te.) Fom t h e above t a b l e, t h e f s t column s h o w s t h e f a v o u t e p o s t o n s,. e. f o m most f a v o u t e s t o longshots. T h e numbe o f a c e s n t h e second column a e d f f e e n t because t h e numbe o f h o s e s n d f f e e n t a c e s s n o t t h e same. In t h s d a t a s e t, t h e mnmum numbe o f h o s e s s 4. The the exstence of a s t a t s t c s n t h e l a s t column suggests f avoute-longshot b a s 3,. e. t h e bettos

44 undebet favoutes and ovebet longshots. Some weaknesses of Al's method ae suggested : 3.2. Bet factons n All's analyss ae teated as ndependent acoss hoses. The constant that bet factons must sum to one ntoduces a necessay but neglected negatve coelaton between bet f actons Al's method depends on the classfcaton (.e. the favoutsm). Howeve, 6th favoute n 6-hose-aces may be qute dffeent fom a 6th favoute n a 0-hose-aces and thus the aveage bet factons may not be vey meanngful When usng the Z-test, Al teated the aveage bet facton as a constant value athe than a andom vaable. We thnk that a andom vaable may be moe appopate hee snce P⑴ s essentally an estmato of the mean subjectve pobablty fo the th favoute hose n ace Moeove, nomal appoxmatons a e not vey accuate when t h e popotons a e v e y small o t h e numbe o f a c e s a e not lage. Hence, Al's Z - t e s t may not b e vald. (One smple w a y t o mpove Al's method s smply t o f n d t h e d f f e e n c e between Y u and P and then t e s t whethe t h e mean d f f e e n c e equals u z e o. Howeve, t h e esultng Z-value would not b e v e y d f f e e n t. ) Al used t h e Bnomal dstbuton f o t h e wnnng event of a 32

45 hose. To fully utlze the nfomaton, the Multnomal dstbuton should be bette because thee ae moe than two hoses. 3.3 Classes of Multnomal Logt Models We eque a model to f t the tue objectve pobablty on the subjectve pobablty of wnnng (.e. the bet facton) (See Atchson (986) o L (986) fo efeences). We stat by assumng that the wnnng pobablty follows the followng fom ln(7 () /tt (n) (omttng the subscpt fo ace ) : ) = a + p ln(p I () /P (n) ) =,2,...,n- (3.9) whee P, () 2 之 P (n) and n ( ) s assocated wth P ( ).e. P,, = bet f a c t o n f o f s t favou Ite and () P (n) = bet f a c t o n f o longshot, e t c. We call ths the A-class snce the logt tansfom, ln(p⑴/ P (n) ), s commonly used by Atchson (986). In ths class of model, the logt (n the multvaate case) of the wnnng pobablty s lnealy elated to the logt of the bet factons. Note howeve, that the esults ae dependent on the abtay choce of the dvso (hose n n (3.9) above) hose. 33 snce all measues ae elatve to that

46 Anothe smla class s exp( a + tt⑴ = I exp( a s In P⑴ + j3 S s (3.0) In Pf J (s) =l 2 " n (wth a n = 0) We c a l l (3.0) t h e L - c l a s s. I t s t h e one o f t h e common f o m s o f t h e Polytomous model (see, f o example, Hosme and Lemeshow (989)) o t h e Multnomal logt model (see fo example. Judge Gffths, H l l, Lutkepohl Sc Lee Both c l a s s e s (985)). a e m u l t va a t e e x t e n s o n s o f t h e bnay l o g t model, b u t t h e A-class depends o n w h c h f a v o u t e h o s e s chosen a s t h e dvso u n l e s s = 0 V. Thus t h e L - c l a s s h a s a n advantage o f s y m m e t y o f paametes s n c e n o dvso s equed. Both t h e A一class and L-class ae qute complcated and we beleve that a smple model should also f t the data qute well. Fo estmaton, we assume : Y 一 = (Y,...,Yn whee Y )T follows a MultnomaK 个) dstbuton s defned n (3.5) and = (tt,..., The lkelhood functon s - I, )T

47 and the log lkelhood can be wtten as = 乙 In tt⑴^ (3.2) =l whee denotes the wnnng pobablty assocated wth the wnnng hose n ace. Maxmum Lkelhood Estmaton can be easly appled to maxmze (3.2) wth espect to the paametes. Notce that when pobabltes equal 0 and bet, factons.e. Ths mples wnnng sk neutalty (See Al(977)). 3.4 Selecton of models Both t h e A-class and t h e L - c l a s s a e e x a m n e d usng some lage d a t a s e t s. Ou hypotheses nvolve t e s t n g w h e t h e and ae z e o s, constants o not. We u s e t h e 6, 7 and 8 - h o s e - a c e s o f Al's d a t a f o t h e pupose o f s e l e c t o n o f models because t h e y nclude q u t e a l o t o f aces. To k e e p t h e analyss smple, w e smply pck t h e a c e s w t h t h e same numbe o f hoses* The t o t a l numbe o f a c e s f o 6 7 and 8 - h o s e - a c e s a e 807, 3532 and 5450 e s p e c t v e l y. The e s u l t s f o A-class a e s h o w n n Fg. 3., 3. 2 and 3.3. And t h e e s u l t s f o L-class a e shown n Fg. 3. 4, 3. 5 and 3.6, The d e t a l e d estmated paametes f o d f f e e n t models a e epoted n 35

48 Appendx A. In these fgues, models ae aanged accodng to complexty. The top box s the smplest model and the lowest box s the most complcated model. Fo example, 0, nsde the top box n each fgue means a = 0 and p = V. Models wth constant a (^0) n the L-class ae gnoed snce ths mples settng one hose apat. The negatve numbes nsde the boxes ae the assocated log lkelhoods and thus lkelhood ato tests can be used. The numbes on the lnes ae the assocated p-values fo the tests that the next models ae bette than the pevous ones. Anothe set of p-values ae assocated wth the tests that the bottom model (quadatc whch wll be explaned below) s bette. Akake Infomaton Ctea (AIC) values ae gven n paentheses below the log lkelhoods. Apat fom fttng the above models, we also ty to epaametze the models as follows : B I = o c + = =a + and (3.3) n ode to educe the numbe of paametes and to captue the tend of ^and oy The detaled esults ae also epoted n Appendx A. Quadatc models ae also ted, but the addtonal paamete s qute sgnfcant n 8-hose-aces only. The followlng quadatc models ae ftted and the esults ae also shown n Fg- 3.-3,6. 36

49 A-class ln(7 (An)) =,2,...,n- L-class : exp[ a + C T = ⑴ ^ exp[ a s s In P ⑴ + 3 J + v (In P⑴ 2 In P s + y (In P (s) (s) 2 =l,2 n (wth a = 0) n Futhe, to see how much bette the estmaton wll be f a much moe complcated logt model s consdeed, the A-class model s extended to the followng one : InCt /tt () n- (n) ) = a + E p InCP / P j=l IJ (j) (n) =,2,...,n- (3.4) t h a t s, t h e wnnng pobablty o f a hose not only depends o n t h e assocat ed bet f a c t o n o f t h a t hose but also depends on t h e b e t f a c t o n s o f a l l t h e othe hoses. Model (3.4) can be eaanged a s follows : 37

50 exp[ a K U T () = ~ n- Ee x P [ + Z 3 J j= InCP j / P ) ] (j) (n) n- a, S + S 8 J= ln(p., / P, J + s J exp a + S ^ E exp[ a (j ) p j= ^ j + S p n) In P In P ( b y p u t t n g p =- S B ) In exp[ a ^ + E 0 j=l j In P J= j I (3-5} T e x pk [ a ^ whee a = a - a n s + I p j=' s j In P (j) and / 3 = / 3 j We may note that model (3.5) s actually t h e extended f o m o f L - c l a s s model n (3.0). Hence, model (3.4) s a n extenson of both A-class and L-class models. Though w e f t t h s complcated model on a v e y l a g e data s e t (5450 S-hose-aces), t h e addtonal paametes ae not sgnfcant at smple model 5% sgnfcance l e v e l w h e n compaed t o t h e and o f A-class model. The assocated esult Is also shown n Table 3.3 and 3.6. Wthn these classes of models, fom the popetes of the models and the esults obtaned, constant-p model ( L e. = 38 0 we pefe what we call the and = P V ). In t h e s e

51 ccumstances, both the A and L classes educe to the same constat-0 model. The easons we pefe ths ae : 3.4. Ths smple model s not affected by the choce of dvso (see (3.9)). Hence, the model s ndependent of the classfcaton Its estmated value dffeent ace-szes. (The s qute consstent n aces wth 入.426,.656 and.35 espectvely.) P's ae sgnfcantly dffeent fom one at 5又 sgnfcance level n all cases hee. Also, the educton of log lkelhood caused by constant-^ n the model fttng pocess s geat n geneal. In addton, usng the Akake Inf omaton Cteon (AIC) (.e. takng the numbe of estmated paametes nto account e.g. Sakamoto, Ishguo & Ktagawa (986)), the constant-p model attans the mnmum values among all the ftted models on both 6-hose-aces and 7-hose-aces n both A-class and L-class. Wth 8-hose-aces, models wth moe paametes acheve slghtly bette AIC values. The constant-p model s as follows : Infx /TT, = )3 l n ( P, ( ) (n) () /P/ ) w h c h c a n b e eaanged a s : 39 (n) = (3.6)

52 P(Y= J P fo =,2,. (3.7) Note that thee s no equement of any odeng conventon. Let n and P be the geometc means of the and espectvely, then l/n = = n j= L S P l/n z thus. Pa o, In In PNP It may be of nteest to notce that the left and ght sdes can be expessed n tems of log atos but the denomnatos ae geometc means. p can be thought of as measung the "Bas" (not exactly the statstcal bas) of the subjectve pobablty wth espect to the objectve pobablty. We would say that the subjectve pobabltes ae unbased f 3 / s not sgnfcantly dffeent fom one,.e. whch Is the sk neutal hypothess tested by many eseaches. If If favoutes and ovebet longshots (I.e. sk pef eence). By usng ths model fo analyss, the weaknesses n secton 3.2 dsappea 40

53 Apat f om usng lkelhood ato tests, checkng the goodness-of-ft Is not easy n ths stuaton. The eason s that the hoses ae not the same n dffeent aces. One smple way s to goup up the hoses accodng to favoutsm and compae the bet factons, estmated pobabltes usng the constant-^ model and the aveage wnnng fequences. In Fgs , each pont s an aveage ove the 500 aces selected accodng to the sequence of the data set. The followng symbols ae used : "plus" : elatve wnnng fequency, "damonds" constant-p pobablty, "tangles" : A s mplct estmates (explaned n model estmated wnnng 3.6.). These gaphs show t h a t t h e constant-p models ae bette than t h e b e t f a c t o n s by educng t h e favoute-longshot bas. Othe models whch ae easonably good a e : Though t h e paamete d and (defned In (3.3)) Is hghly sgnfcant In 8 - h o s e - a c e s, t s not s g n f c a nt n t h e 6 & 7-hose-aces. Smlaly f o oy Ths may b e due t o t h e lage amount o f data n 8 - h o s e - a c e s. Nevetheless, w e beleve t h a t t h e c o n s t a n t ^ model captues most of t h e vaaton (as shown by t s e f f e c t on educton of log lkelhood) and due t o t s smplcty and consstency n d f f e e n t pats of data s e t, w e w l l extend t h s smple dea f o moe complcated b e t s n Chapte f o u. Now, w e consde t w o popetes o f t h e constant-p model a s follows 4

54 Popety 3, Suppose 卩 > L We defne whee then Poof Consde 3, P 3 " >lw A j P^" J P ^ d - P )q p. Smlaly f o ^, E 42

55 8- j That means, when P! Is geate "than a weghted aveage of othe and the evese condton holds fo. In patcula, when s the most favoute hose,.e. P > P V Jj^, j S Also, when s t h e longshot hose, Le. V P snce. Popety 3.2 s nceasng n S j f In > (In P)3* s deceasng n 3 f In < (In P) whee E w In P (In P) whee w. E w. Poof P p In P mples (In (J P, > ^ P f In P w h c h mples that ^ d d 7 ^ ^ (T F ^ P 3 In P ^ -(T P 43 I - P 0 P ^ In F 2 ^ <

56 Smlaly fo That s, f In d 0 <0 f In P < (Inp ). 圔 Is geate than a weghted aveage of In P s nceasng n p. Smlaly f o t h e condton t h a t u s deceasng n p. I t can. b e ntepeted a s t h a t when p Inceases, wnnng pobabltes o f moe favoute hoses ncease but t h a t of l e s s f a v o u t e hoses decease.

57 Fg. 3. A - c l a s s models f o 6 - h o s e - a c e s (no. o f a c e s = 807) ^ ( ) 807,.2363] / ( ) [806,.823] ( ) [802,.0922] ( ) [805,.583] ( ) [ 8 0 2, (2469 4) [80 347] L e ge n d ( ) [80,.656].304 l o g k.64 ( ) ; A I C p -v a l u e fo the t e s t t h a t the bottom model s bette Numbe s h o w n o n l n e : p-vau e f o the t e s t that n e x t one s b e t t e a, js, ( ) [797] ( ) [

58 Fg class models f o 7-hose-aces (no, o f a c e s = 3532) 0, (566.70) [ ] (55.56) (532.2] [353, [353,.000 ] /.466> \ ^ (55.34) (549.74] [3526 [3526, ] (538.28) (539,70) [3525, [3525, 255].999] - v e v a l u e s :l o g l l k ):AI C f o t h e t e s t t h a t a, J I t h e b o t t o mm o d e l s b e t t e (547.48) Numbe s h o w n o n l n e p - v a u e f o : [3520 t h e t e s t t h a t n e x t o n e I s b e t t e.0525, (

59 Fg. 3.3 A-class models f o 8-hose-aces (no. of aces = 5450) ( ) [5450,0 o,p ( ) ( ) [5449,0] [5449, o,3 l ( ) ( ) ( ) [5443,0] [5448,0 [5443,.0002] ( ) ( ) [5442, [5442,.025].000] 029< Leg end : ( ) ( ) [5394] [5436] ( ) [5435] - v e value s :l o g l l k () AIC I» I:d. f., p- v a I ue f o t h e t e s t that t h e b o t t o m model s bette Numbe shown on ne : p- va uc f o the test t h a t n e x t o n e s bette

60 Fg. 3.4 L-class models f o 6-hose-aces (no. of aces = 807) (2465,48) [807,.0380],078 0, gnoed (246.86) 806,.304] gnoed ( ) ( ) [802,.0086] [80, gnoed (2469.4) [80,.0347] 0209 fo t he te the bottom that ( ) [796] s bette Numbe the.5839 that ( ) [795]

61 Fg. 3.5 L-class models fo 7-hose-aces ( n o. o f a c e s = 3532).000 0, (532.2 [353,.8827].509 0, (55.34 [3526, 0348 gnoed [3525,.954, (538.28) [3525, 986] Le g,.976 a.i C : ogi fo the t e s t t h a t t h e bo om model s bette Numb s s h o w n o n l n e :.u e f o the t e s t that n e x t one s bette (549.OS) [359] (550.76) [358] 49

62 Fg. L-class models f o 8-hose-aces (no. of aces = 5450) ( ) [5450, gnoed ( ) [5449,.000] gnoed ( ) (53650,0) [5443,0] [5442,.939] gnoed ( ) [5442,.098] 306( ( ) (53657,72) [5394] [5435] ogl!k ):AIC.029 f o t h e t e s t that t b o t t o m model s bette ( } [5434] the b e t te 50

63 Fgue 3, 7 Fst fcvoultoe (8 hcbe occs) fc^t facton <av^ wn f^q 办 b«t facton 0,34 ptob, m Fgue 3. 8 Second fowutob (8 hoeo <K»s) ^ ^ 一"b^k factbn + 0,^ b^k facton o/ 5 6 -JU_f. t 2m a2 0.2 CX2I3 赠 紘25

64 Fgue 3, 9 S«vwth fcp/out«s (8 ho «ac«s) o.os , 毳 ,038 o 0,036 A <> , , b^t facton + w e wn f^q o 一 b^t facton wtlmot^d pob, 厶 Fgue 3, 0 E<jhth fewltw (8-hon»-oc#«) , , , ,022 0,02 0, , ,02 0,022 一 b«t facton ,026 fcktk?n w e wn f^q 52 0M B wtlnx^k pob 么 对 么 032

65 3.5 Empcal analyses n the U.S H.K Japan and Shangha We have ftted the constant-j3 model n dffeent acetacks In the U.S- Inteestngly, the estmated p anges fom.02 to.56 n all acetacks and all of the p's ae geate than one at % 5 level of sgnfcance. If we consde a two-sded test, we conclude that all of the p s ae s g n f c a n t l y d f f e e n t fom, one a t 57. level e x c e p t n Atlantc Cty. To s e e w h e t h e t h e same s k p e f e e n c e behavou appeas n t h e othe acetacks, w e t e s t w h e t h e p s d f f e e n t f o m one and w e also t e s t w h e t h e p s geate than o n e s e p a a t e l y. In Japan, t h e p Is also s g n f c a n t l y geate t h a n (8c d f f e e n t f o m ) one though t h e estmated value o f 尽 s smalle than those of the U.S.. Howeve, n Hong Kong and Shangha data sets, the js's ae not sgnfcantly geate than one at any easonable level (a esult consstent wth Busche aces n Hong Kong. S c Hall (988)) despte havng lage nunbes of In Shatn, the p s sgnfcantly dffeent fom 八 one due to the smalle value of 择.Howeve, oveall, all the estmated p*s ae close to one and thus the bet factos should be easonable estmates of objectve pobabltes. esults ae summazed n table 3.2 (a) and (b). 53 The empcal

66 Tab I e 3. 2 Summay of p-model f o w n bet n d f f e e n t a c e t a c k s Racetack ^ t a c,e s 03 / As e/> O ),, p^- v a l u e p - v a l u e la) Ibj Aveage pool 士 s z e m( U S二 $ ) Hong K 0 n g : Happy V a l l e y 22 2 (8-89 ) Shat n 943 (8-89 ) S h a t n 455 Happy V a l e y (8-89 ) Al's »083» # »088» # I, 0 8 5, # (70-74) Saatoga ^ 0 «0 24,79 0* Roose v t ! ^ 0 ^ 0 28,2 8* Yonkes ^ 0 «0 228,5 5* unknown Q u a n d t ' s (83-84 A a n t c Cty M e a d o w a n d s Japane s e (90) Chna , ,9 2 9, unknown : Shangha ( ) Note f o Table 3.2 : () *Ths amount s t h e aveage total pool s z e Incudmg a l l pools. (I) # t h e amount may not be meanngful 54 due t o t h e e f f e c t o f

67 nflaton dung The aveage pool sze In 989 was US$,996,420. () All the aveage pool szes ae n U.S. dollas; the exchange ates used ae : US$ = Y 37.8, US$ = 7.8 HK$. (v) p-value (a) Is assocated wth the followng pa of hypotheses H o : = and : based on t h e lkelhood ato test; and p - v a l u e (b) s assocated w t h t h e followng pa of hypotheses : H 八 A 八 :0= and H : p>l based on the test statstc ; O-D/seO). 3.6 Compason wth the othe models Pevously, we have suggested the constant-3 model whch s smple and easonably good. In fact, thee wee some othe smla models ethe mplctly o explctly mentoned by othe eseaches Al's mplct estmates of wnnng pobabltes : Al (977) studed the elatonshp between utlty and wealth level epesented by the etun of bettng a hose as explaned below. As befoe, let odds on hose. In addton, f a betto bet M dollas on hose t 55

68 odds on hose. In addton, f a betto bet M dollas on hose, hs expected utlty on ths hose s EOJ) = (l-7)u(xo-m) + 7U[(Xo-M) + Md+O^l (3.8) w h e e X o = t h e ntal captal o f t h e betto, U(.) = h s u t lt y functon of wealth. Assumng t h e betto Is an expected u t l t y maxmze, he must be n d f f e e n t between bettng on any t w o of a l l t h e hoses In a ace,.e. E(U) = ECU ) = = E ( U ) 2 n (3.9) whee n s the numbe of hoses n that ace- Now, eaange the ode of hoses by favoutsm. I.e. and assume futhe that X - M = 0, UlMtl+O^)] = M UCl+O^), U(0) = 0 and U(l+0 )=LFom (3.8) and (3,9), n U ( l + 0 ) = 7 /, n =l,2 ",nki (3.20) Usng all t h e 8-hose-aces o f Al's data s e t, Al computed t h e weghted aveages of a l l t h e ealzed odds ( f o detals, s e e Al (977)) and appoxmated t h e wnnng pobabltes o f hoses (会夏 In dffeent favoute postons by the elatve fequences of wnnng. 八 He obtaned estmates of the utlty fnctlon by substtutng and O^nto (3.20) and then t h e f o l l o w n g elatonshp w a s f t t e d by t h e l e a s t squaes method 56

69 log n U(x) = log (x) 0 0 (3.2} R 2 = (The estmated ntecept tem was not epoted coectly In hs pape.) 0 U(x) = x L (3.22} w h e e x = wealth = +0^ f o any hose L Snce U'(x) > 0 and U (x) > 0, t h e absolute sk-aveson measue, -U (x)/u (x) nceases w t h wealth, Al concluded t h e epesentatve b e t t o t a k e s moe sk a s h s captal declnes. Theefoe, bettos a e s k - l o v e s n geneal. Usng t h e u t l t y and w e a l t h elatonshp n (3.22), w e can e s t m a t e t h e wnnng pobabltes f o each a c e : ⑶ ⑴ 八 7 = whee 0! ) (X23) E /u(+o s ) u s Ud+O^} ae computed usng (3.22) and the actual odds. To compae these estmates wth ou smple constant-p model, we smply compute the log lkelhood and the esult s shown In Table 3.3, 57

70 Table 3.3 Compason among dffeent methods of estmaton Method l o g l k Upaametes Wn bet f a c t ons Const ant-p model A l ' s estmates By dectly compang the log lkelhood values, ou constant-p model s slghtly bette than the model mpled by Al's utlty functon. We also show the compason gaphcally n Fgues The gaphcal compason s not vey clea whethe ou constant-/3 model s bette snce both estmates ae qute close to the aveage elatve wnnng fequences. A fomal test based on log lkelhood values s moe useful n ths case. Cox's test (see Cox (96,62)) s caed out to test whethe ou constant-/? model s bette than Al's estmates, Le, to test : : constant-p model vs. : The test nvolves two stages. Fst, evaluate : (loglk f - l o g l l k - E ( l o g l k f - logk SD(loglkf - logllk^ Hf) 58 j Hf)

71 then, evaluate (loglk X - loglk ) - EClogllk - logllk! H ) g SDdoglk s whee logll^ = the logllk unde g 8 g wth the estmated paamete a=f,g), E(. H^) and SD(. ae expectaton and standad devaton when s tue. Unde Hf, T «N(0,). Unde H, T - N(0,). f s g If H 容 s bette than H f, T f wll be lagely negatve. If H s bette than H, T wll be lagely negatve, f g g Often, t s dffcult to evaluate the expectatons and the standad devatons. Some authos (e.g. L(989)) popose usng the bootstap method (Efon(982)). Ou pocedue goes as follows. Step () : Ft the constant-3 model and Al's utlty functon usng the eal data (Y 一 ", Y m) whee YT denotes t h e vecto o f wmlog event o f a c e. Obtan loglk and logllk Denote t h e s e t w o models f g by M and M espectvely coespondng t o H and H. f g f g Step () : Fo k = (k) (a) Unde Mf, obtan smulated data (Y一 " 丫 ) and then obtan m t h e paametes assocated w t h t h e t w o models, loglk (k) g. Save t h e last t w o values. 59 loglk( k) and

72 (b) Unde M, obtan smulated data (Y Y 产 a n d then obtan g m t h e paametes assocated w t h t h e t w o models, loglk (lc) z loglk^k5 and Save t h e l a s t t w o values. Step () Compute t h e bootstapped expectatons and standad devatons unde H^and b y usng t h e values obtaned n Step (). In ou case, B = 00 Is used and t h e f o l l o w n g t e s t s t a t s t c s a e obtaned T = 3.052, f T = g T h e e f o e, w e conclude t h a t ou constant-g model s b e t t e a t any easonable l e v e l o f s g n f c a n c e. We n o w nvestgate t h e d f f e e n c e b e t w e e n ou c o n s t a n t ^ model and Al's u t l t y f u n c t o n. Unde t h e u t l t y maxmzaton assumpton, ECU^) = tt^uq+o^) = K (a postve constant) V and hence, usng ou constant-p model, w e have : U ( + 0) = = 一 石 f o 置=,2,"., o. log^ud+o^ = l o g o K + s (3.24)

73 Snce K s any postve constant, we can fo smplcty take K =. Also, p /( + o ) Cl-t)/(l + o ) = L = L - (-t}/( +o ) E /( + 0 ) Z (l-t)/(l + O ) s s s s whee t = tack take. (Note t h a t f Beakage s absent, the above ^ should b e =.) Then, fom. (3,24), w e have : log UQ+O ) ^ l o g K U 3 l o g (-t) + log U U U X ( P s S + 3 logqa+oj (3.25) T h e e f o e, compang (3-2) and (3.25), the d f f e e n c e of the two methods s manly due t o t h e f a c t that n (3.25), t h e ntecept tem s not a constant and depends on a patcula ace AMQ's logt model : Asch, Malkel and Quandt (984) f t t e d a logt model (heeafte called t h e AMQ logt model) f o t h e elatonshp between t h e t u e wnnng pobablty and t h e b e t facton. Ths Is smla t o but not t h e same a s ou constant-p model, AMQ^s logt model s a s f o l l o w s : 7 = expf^p^ I exp(<p ) S 6 whee s a paamete

74 (Note that n the ognal pape, the paamete Is p, we change ths to Usng the Atlantc Cty data (72 aces), we compae the constant-p model wth the AMQ logt model. The esults ae shown n Table 3.5. Tab l e 3.5 Compa I s ons among d f f e e n t model Method logl k est W n bet factons C o n s t a n t - p model AMQ l o g t model A paamete 尽= = (Note that ou esult fo AMQ logt model s vey close to but not exactly the same as the esult obtaned n the ognal pape (log lkelhood = ). It s because they used : P. ^ +0 ) so E P *. J J Results epoted hee use (3-4) so Z =.) Hence, the AMQ logt model s much wose than model and n fact, the smplest model woks much bette than the model. Note that n ou model, when p=l, t educes to the smplest model but In the model, t cannot be educed to ths smplest model by estctng the paamete q. 62

75 Cox s t e s t s agan caed out t o t e s t w h e t h e t h e smplest model s b e t t e than t h e AMQ logt model,. e. t o t e s t : 7, = v s. 义 A M Q logt model Agan, t h e bootstapped Cox's t e s t Is used and t h e numbe o f bootstapped samples, B = 00. The f o l l o w n g values o f test s t a t s t c s a e obtaned T = , f T = g T h e e f o e, w e e j e c t AMQ's logt model n f a v o u of t h e model a t any easonable level o f sgnfcance. The constant-p model s a t l e a s t a s good a s t h e model 7^= by t h e lkelhood a t o t e s t. 3.7 R e l a t o n s h p b e t w e e n t h e b e t a and pool s z e s Recall t h a t /3 measues t h e ba s o f t h e b e t f a c t o n s w t h e s p e c t t o t h e t u e wnnng pobabltes. We suspect t h a t f t h e pool s z e o f a ace s hghe, t w l l a t t a c t moe pofessonal b e t t o s t o bet and thus, t h e longshot-favoute ba s w l l b e educed,. e. p w l l decease (o appoach t o ). T h s dea h a s also been mentoned n Hausch, Zemba and Rubnsten (98} : "Intuton suggests t h a t Santa Anta moe accuate estmate o f 63 w t h t s lage bettng pools would have t h e wnnng pobabltes t h a n would be

76 obtaned at Exhbton Pak." To test ths hypothess, the followng model s ftted H v ^ 3 (3.26) whee the addtonal subscpt denotes the ace numbe and 0 s defned as follows. Let W = pool sze (Le. total wn bet) fo ace. () / 3 = + e/w (As W () > oo, /3 > ; a s W B ( > 0, p > oo, p > ; as W ~ > 0, (As W > 3 > as W > B I = a/w b J3 > +^.) (3.29) p = C + e/w (v) > o (As W () 3,27) (As W > 0), B > 0; as W ^ 0, p > oo ) (3.30) 0, J5 > m whee e, postve, t h e above appoachng condtons w l l be s a t s f e d. The MLE f o t h e s e e x t a paametes can be f o u n d easly. Empcal e s ul ts f o Meadowlands ae shown n Table 3.5. (In t h s secton, w e adopt t h e

77 nave way of compang the log lkelhoods dectly.) T a b l e 3.5 Poposed models f o the e l a t o n s h p between p n Meadowlands (705 a c e s ) Model tt paamet e s est mates & W log k = (3.27) = (3.28) ^= ]= (3.29) ^= e=24264 (3.30) a = l b=0.455 Among the fou poposed models, (3.27), (3.29) and (3.30) ae qute good. It s because (3.29) and (3.30) ae sgnfcantly bette than constant sgnfcant than (3,27). Ou am n ths secton s manly to test the hypothess that the sk behavou s elated to the pool sze of a patcula acetack. Fom ou analyss In Meadowlands, ou data appeas to suppot ths. Howeve, empcal esults n Japan and Hong Kong ae not consstent wth Meadowlands and no systematc dependence on pool 65

78 sze can be found. The empcal esults ae shown n Table 3.6 and Table 3.7 fo Japan and Hong Kong, espectvely (all the estmated paametes ae adjusted such that the pool szes ae n U.S. dollas). Note that the estmated paametes ae not qute consstent n the thee data sets. In addton, n some cases, even the sgns of the paametes ae not sensble (.e. not postve). Table 3.6 Poposed models f o the e l a t o n s h p between n Hong Kong ( , Model paametes 874 a c e s ) est mates log I k /3 = (3.27) = (3.28) ^= ? = (3.29) = e= (3.30) a=l b=l

79 T a b l e 3.7 Poposed model s f o the e l a t o n s h p between & W! n J a p a n (607 a c e s ) Model /3 = S I = # paametes est mates 0 3 l o g k p=l (3.27) c = (3.28) 2 ^= ^= (3.29) 2 = l e = (3.30) 2 a=l b= Theefoe, t h e dea of elatonshp between b e t a and pool s z e only succeeds n Meadowlands. Ths may b e because : () No elatonshp e x s t s. The sgnfcance of t h e elatonshp n Meadowlands s due t o some othe easons, o () The elatonshp depends on acetack, cultue o some specal easons. Note that t h e favoute-longshot bas n Meadowlands Is cleae snce we consde hee, o

80 () The elatonshp may be appoxmately tue fo the U.S. acetack but the ef feet may be tval when the pool sze s as bg as n Hong Kong. That s, no fm concluson can be dawn fo ths elatonshp unless A we can have data fom moe tacks wth lage 3 ( o small pool sze. 3.8 Monng lne odds and eale odds In the pevous sectons, we have checked the systematc bas of the bet factons. The bet factons ae obtaned just befoe the ace stats, the assocated odds ae called fnal odds. In addton, thee ae monng lne odds (n U.S.) and eale odds. Befoe the stat of a bettng peod, the offcal handcappe gves estmates of the wnnng pobabltes fo each hose whch ae expessed n odds. These ae called the monng lne odds* These odds ae detemned by the judgment of the handcappe based on such factos as a hose past pefomance, dstance o f t h e ace, weght caed b y t h e hose, and past pefomance o f t h e jockey, etc.. We would lke t o know whethe t h e s e odds ae bette than t h e f n a l odds. Moeove, t h e odds detemned b y t h e bettos a e changng ove t m e u n t l t h e s t a t o f a ace. We have obtaned some odds data b e f o e t h e s t a t of a ace. It Is nteestng t o s e e whethe t h e f n a l odds a e bette than t h e pevous odds n estmatng t h e 68

81 wnnng pobabltes of hoses. We have obtaned estmates of 3 J n Atlantc Cty (72 aces) usng the above two knds of odds and the empcal esults ae shown n Table 3.8. To compae the monng lne odds and eale odds wth the fnal odds. C o x t e s t s appled and t h e esults ae shown n Table 3.9. In t h s case, analytcally bootstappng s not necessay because w e can obtan the condtonal devatons easly. To t e s t H expectatons and standad vs. H, t h e t e s t statstcs T f g f and T a e computed a s b e f o e and t h e condtonal expectatons and z standad devatons a e gven by : fa f EUoglk, - logllk J = E Z ^ ( l n- ^ } VCloglk - l o g l k H fa Va(Y I In fa fa V(Y I H) = f Cov(Y t, Y I s Cov(Y,Y f H ) In E E Un s ^s fa 7 fa (- fa -Tt 69 fa U s

82 Smlaly fo Edoglk - loglk 5I H ) and g f g Vdoglk - loglk I H ) f g whee loglk = log lkelhood value f o t h e model unde H ^ k (k=f,g), m I In = 0 o accodng t o whethe hose w n s t h e a c e k八. = estmated wnnng pobablty of hose n the ace unde H k (k = f,g). Table 3. 8 八 Compasons of p 's u s n g d f f e e n t odds odds t y p e seo ) 卩 Monng l n e odds loglk(l) logl k ( ) Odd s a t t m e b e f o e a c e s t a t s 3 m n mn mn mn mn mn F n a l odds 70

83 T a b l e 3.9 C o x ' s tests f o comp a n g f n a l odds w t h othe odds F n a odds M o n n g l n e odds Odds a t F n a l odds ,3762 t m e befoe F n a l odds F n a l odds F n a l odds F n a l odds F n a l odds m n. Fom Table 3.8 and 3.9, t s c l e a t h a t, t h e f n a l odds a e much b e t t e t h a n t h e monng l n e odds. Also, t h e monng lne odds A h a v e a moe s e o u s f a v o u t e - l o n g s h o t b a s (lage 卩)than the fnal odds. Ths s not vey supsng snce the fnal odds ae detemned by a vey lage numbe of bettos but the monng lne odds ae detemned by a few people only. Ths esult s qute consstent wth Asch, Malkel & Quandt (982,984). Futhe, the log lkelhoods, loglk(l), f o the odds collected befoe the ace stats ae nceasng ove tme. Also, Cox s t e s t s suppot t h a t t h e f n a l odds a e b e t t e t h a n a n y e a l e odds above a t 5% l e v e l o f s g n f c a n c e. A s e n s b l e explanaton s t h a t t h e b e t t o s w h o ha ve some nsde nfomaton p e f e t o b e t a s l a t e a s possble n o d e t o e d u c e t h e t m e t h a t t h s sgnal s avalable t o t h e o t h e 7

84 bettos. 3.9 Conclusons 3.9. Afte gong though a sophstcated modellng pocess, the constant-^ model seems to be easonably good. Also, t s easy to ntepet the / On fttng a constant-p model n dffeent acetacks, all the estmated values of 3 / ae qute close to one. Futhe n some acetacks [3 s s g n f c a n t l y g e a t e than one a t 5% level. Ths suppots t h e tadtonal vew of a favoute-longshot bas. In othe acetacks, howeve, bas cannot b e f o u n d The constant-p model appeas t o b e slghtly b e t t e than Al's mplct e s t m a t e s o f wnnng pobabltes. The AMQ logt model s even w o s e t h a n usng t h e b e t f a c t o n s dectly The elatonshp b e t w e e n pool s z e and favoute-longshot b a s appeas t o e x s t n Meadowlands. Howeve n o t e t h a t, b y compang t h e values of amd does not e x s t n Hong Kong. 72

85 3.9.5 The fnal odds appea to be sgnfcantly bette than the monng lne odds n estmatng the wnnng odds and eale pobabltes. 73

86 CHAPTER FOUR ANALYSES OF MORE COMPLICATED BETS 4. Intoducton In ths chapte, we wll epot the analyses of some moe complcated hose-acng bets 一 exacta, tfecta, qunella, double bet and double qunella. Havlle (973) Heney (98) and Sten. (990) poposed t h e e d f f e e n t models f o estmatng t h e moe complcated pobabltes but no detaled empcal studes can b e f o u n d. We u s e logt models t o compae t h e s e models. Moeove, data based on d f f e e n t b e t types c a n also b e used t o estmate t h e s e complcated pobabltes. Thus, compasons a e made among both models and b e t types. All t h e conclusons w l l be pesented n t h e l a s t secton. 4.2 Descpton of some poposed models 4,2, H a v l l e m o d e l The smplest and most commonly used model t o estmate t h e complcated pobabltes s t h e one poposed b y Havlle (973). The basc d e a s smple. Fo nstance, t o pedct FChose I wns and hose j f n s h e s 2nd), w e may smply u s e : 74

87 7 (4.) j f and ae known C n^s can be estmated by bet factons ) A s m l a d e a w a s also mentoned n Plackett (975), Moeove, t s the ankng model poposed by Luce Se Suppes (965} n t h e study o f choce behavou. At a f s t glance, (4.) may seem easonable and thus, some e s e a c h e s used t h s method f o estmatng pobabltes (e.g. Hausch, Zemba & Rubnsten (98) ). It s also known t h a t bettos u s e t h s method n pactce. some Howeve, tt小(.e. P(hose j fnshes second hose wns) ) may not be equal to ^/d-n^) n geneal. One common agument s mentoned n Hausch, Zemba and Rubnsten (98) : "no account s made of the possblty of the Slky Sullvan poblem t h a t s, some ho se s geneally e t h e w n o f n s h o u t - o f - t h e - m o n e y ; f o t h e s e hoses t h e f o m u l a s g e a t l y o v e - e s t m a t e t h e t u e pobablty o f f n s h n g second o thd". One easonable w a y t o f n d t h e s e complcated pobabltes s t o assume a n undelyng pobablty dstbuton f o t h e unnng t m e s o f hoses. I t can b e s h o w n t h a t f t h e unnng t m e s f o l l o w e x p o n e n t a l dstbuton ndependently w t h d f f e e n t mean unnng t m e s, (4.) w l l b e obtaned. Fo completeness, t h e f o l l o w n g poof s ncluded. L e t T b e t h e unnng t m e o f h o s e, =,..., Suppose 75

88 T f expd/e^ ndependently,.e. = - exp(-t/0^ t^o. Then, w<t» TC =p( V o =J 0 5 l [ - F ( t 0^)] f(t[0^) d t w h e e F { t j0^3 s t h e cdf of T e x p C - t J ^ ) dt /9 f o =l 2 n /0 P ^ < s^!?j {T (4.2} s>) I e x p ( - g ^ l/0 s ) [ - exp(-t/e.) ] dt 3 0 =tc ( S /0 } f exp [-t(z /0 ) ( I-tt ] { l - e x p [ - t ( Z / 0 Jtt ]> d t j = J 0. 7C Tt = whch s (4.). Obvously, t h s f o m u l a also h o l d s f w e assxune t h e m m n g t m e s d s t b u t o n s ndependent Webull w t h common shape and locaton pa a m e te s. Moe g e n e a l e s u l t s can b e f o u n d n Danse (983). 76

89 4.2.2 Heney model Heney (98} agued that a cuous, and pehaps unealstc, featue of the Havlle model was that (4.} dd not depend on the numbe of hoses n the ace. He suggested to assume that the unnng tmes ae ndependent nomal wth unt vaance,.e. NCe^l) ndependently. The esultng pobabltes ae obvously the same as that of a geneal constant vaance model (.e, V(T ) = constant VI). Unde the Heney model, I t t -0 ) dt...dt n 乂 n n (4.3) whee <})( ) s t h e standad nomal pdf Howeve, computng (4.3) s d f f c u l t and even computng s n o t e a s y because, unlke t h e Havlle model, no closed f o m soluton can b e f o u n d. Heney suggested t o u s e a Taylo s e e s expanson about P [T <T2 么P [T <T2 <...<Tn]J<

90 whee Mj. n s the expected value of t h e th standad nomal ode statstc n a sample of sze n. Altenatvely, P [T <T <T ] = $ { " P [T <T <...<T ] > I C + n!0qj -T^T e, :n h C4.5) whee by Taylo's expanson about 0 = f o t h e t e m nsde t h e lage backet. Usng smla methods, P [T s smallest] ^ l / n + 0 u /(n-) I ; n o $ [ z whee z o o f. n ] (n~l}0cz o ) (4.6) (4.7) = $ ^I/n). Hence, by usng (4 6) o (4.7), w e can have estmates o f 0 f t s known o t h e wn bet fpactons ae accuate estmates of n. Then, w e may substtute t h e estmated values of 0 nto appopate equatons t o obtan estmates of moe complcated pobabltes. Fom ou expeence, usng methods smla t o (4.4) and (4.6) poduces a lage numbe of negatve pobabltes. Theefoe, w e concentate on t h e dea of (4.5) and (4.7) n ou analyses. Fo example, 78

91 = P( t, 丁 < othes) ; n +0/ 广2;n (0 + 0 ) (fx + p ) + ~5 ^ ll! ]} n - 2 (4.8) W h e e a a n d y = In pactce t o " satsfy the unt-sum constant, smple scalng s usually necessay. Though t h e t t l e o f Heney (98) nvolves t h e w o d "hose aces", h e dd n o t analyse any hose-acng data by h s method Sten model To e x t e n d t h e Havlle model, a natual choce s t h e Gamma unnng t m e s model poposed by Sten (990). In h s pape, t h e pobablty o f a pemutaton w a s s e t equal t o t h e pobablty t h a t k ndependent gamma andom vaables w t h common shape paamete and d f f e e n t s c a l e paametes a e anked accodng t o t h e pemutaton. Ths dstbuton w a s motvated b y consdeng a competton n whch k p l a y e s, scong ponts accodng t o Independent Posson pocesses, w e e anked accodng t o t h e t m e u n t l pont s w e e scoed. Gamma models can b e used t o e s t m a t e t h e moe complcated pobabltes n h o s e - a c n g w h e n only t h e pobablty o f wnnng w a s gven f o each ho se,. e. m j e p Gammaf, ^ Independently f o = e f e x p O ^ ) 79 t >0. o,

92 whee s pedetemned and can be estmated fom (o P^}. Obvously, when =, the Sten model becomes the Havlle model, and when > oo. It becomes t h e Heney model. Howeve, t h e pocedue f o solvng 0 and t h e fomula f o computng " ae, In geneal, qute complcated. As a vey small empcal study. Sten analysed found t h a t t h e use of =l (whch s t h e Havlle model) f o estmatng complcated pobabltes Is less accuate than that o f =2. Fo all of t h e above models, w e should note t h a t, n f a c t, any monotone nceasng tansf om of the unnng tmes follows ndependent nomal dstbutons mples t h e Heney model and smlaly f o t h e Havlle and Sten models snce => gft^) < g(tj) f g(.) s any monotone nceasng functon E x t e n d e d l o g t model f o moe c o m p l c a t e d b e t s In Chapte thee, w e have suggested t o f t t h e smple constant-^ model n ode t o analyse t h e w n b e t data, Le. InC / I tc ) = p In(P / P ) k k f o any I,k (I^k)

93 whch means the logt (n the multvaate case) of the wnnng pobablty depends on the logt of the bet factons n a vey smple way. Usng a smla model stuctue fo condtonal pobabltes, we have f o w h e e 丌 小 = P ( hose j f n s h e s second hose wns P ( hoses I & j f n s h f s t ) S c second esp.) P ( hose I wns ) (4.9) Hence, we have : (4.0) Z P s lp s h whee P = wn bet facton P 小, P j / p and f ae paametes t o b e estmated. Moe geneally, _ ^ s^ s w h e e t =, and A, n J A = / _..(4J0 s estmated by t h e Heney o Sten model U A 8

94 To estmate n by the Heney model, j we fst estmate 0 by (4.7). Then t w o stateges ae employed : Use (4.8),.e. the appoxmaton fomula poposed by the Heney ⑴ model to estmate u : j () Numecal ntegaton, n patcula Gauss-Hemltan Quadatue, s used. (Fo detals, see Appendx B.) Fo the Sten model, numecal ntegaton s employed. Now, ( 0 (4.2) othewse Assumng Y(2)= (Y,...,Y )T ~ Multnomal (丌⑵) 一 2 n»n-l whe e u ( 2 ) n 補 = [n 丁,...,7 ). We have t h e f o l l o w n g log lkelhood : m = = y y n tt j = T In T, [ tl2]l (4.3) = w h e e t h e subscpt [2] denotes t h e wnnng hose and t h e hose f n s h n g second n ace, and m = total numbe of aces* Hence, maxmum lkelhood estmatos f o t h e paametes can be easly obtaned. Note 82 that 八 and 八 f wll be asymptotcally

95 uncoelated because dl3 dj _ In ths chapte, we manly use the above dea to analyse ou data sets wth dffeent models and dffeent bet types. 4.4 Empcal analyss fo exacta bet In ths pat of study, we choose the exacta bet data In Meadowlands. Exacta bet means the type of bet that the bettos ae equed to guess whch hoses wll fnsh fst and second, espectvely, n exact ode. To estmate the exacta pobablty (.e. h^), the exacta bet facton tself s thought to be a good choce. Anothe choce Is to use the wn bet data but ths Involves estmaton of fom estmates of The Havlle, Heney and Sten models ae thee altenatves fo dong ths task. To facltate the dect compason between the exacta bet factons and the estmated pobabltes estmated by usng dffeent models, the magnal exacta bet facton s also E Defne P " - exacta bet facton fo magnal exacta bet facton E hose I and j, P s defned to be E the P U. Note that unlke t h e smple w n bet, even cm-tack bettos cannot s e e t h e nstant changes of odds of t h e e x a c t a b e t. 83

96 The empcal esults ae shown n Table 4.. Note that, In Table 4., (,) s the log lkelhood value A. f o a patcula model 八 wthout paamete estmates but O,fx) s the log lkelhood value fo a model wth paamete estmates. Fo nstance, when the paamete estmates ae added to the Heney model, the esultng estmated pobabltes ae no longe based on the Heney model. T a bl e 4. Compasons among models f o e s t m a t n g exacta pobabltes B e t type / method (,) ^ E x a c t a bet ^ I , ap p o x almo s t exact Wn bet : Hav e Heney : Sten : = I = I , Magnal exa e t a : Hav e Heney : a l m o s t exa Sten ( N. B. t : : I s t a n d s f o log l k e l h o o d. ) 84

97 In the followng, we manly emphasze on the compasons among the Havlle, Heney, Sten models wthout paamete estmates. It s because the patcula model (such as Heney) wth paamete estmates s consdeed to be complcated to use n pactce and, moe mpotantly, the thee models poposed n lteatue have theoetcal backgounds 一 the patcula unnng tme dstbutons. By dectly compang the log lkelhood values (,), we can see that the exacta bet facton s the best one. Fo both the wn bet facton and the magnal exacta bet facton, the Heney model appeas to be bette than the othes. We may compae the model fts of dffeent models moe fomally based on Cox's tests agan. Recall that Cox's test (Cox(96,962)) nvolves two steps. To test the Heney vesus the Havlle model, the Heney model s assocated wth the null hypothess and the Havlle model s assocated wth the altenatve hypothess. Afte computng the test statstc, the pocedue s evesed,.e. the Havlle model s teated as the model unde the null hypothess and anothe test statstc s computed agan. In geneal, to test two models unde and H the test statstcs ae g. Unde H, f T = (loglk - logllk ) f J -N(0,) and unde H f g f V(logIkf - l o g l k j

98 丁 (loglk - - l o g l k ) - E ( l o g l k - loglk I H ) f g S / / VCloglk «f «~ N(0,) w h e e EC. H^) and V(. H^) a e expectatons and vaances unde loglk k = log lkelhood value f o t h e model unde H k (k=f,g), nk7 = y y y Y j k TC^ 广 estmated pobablty of hoses S c j fnshng fst & second, espectvely, n ace unde fa E( 0 gk f - logkj Hf) - E Z E \ ( l n 7l] j Vdoglk - l o g l k H ) f g fa =厶 E 飞 I 厶 E 乙 E j 2 VaCY tj I H )f +, ga 乂 f f A E j s no t (= Cov(Y \)t,y fa J 7 j, & K I H ) ='fnxxn s, P f j,l U / s, I s, S m l a l y f o EClogUk^ - loglk^j H^) and Vdoglk g - l o g l k I H ) z The e s u l t s o f Cox's t e s t s a e shown In Table J

99 Table 4.2 Cox's tests f o exacta (50 aces). E x a c t a bet Heney Hav I l e Sten f a c t on. E x a c t a bet facton. E x a c t a bet facton. E x a c t a bet facton (=2) Sten (=5).Heney Hav le heney Sten Havlle Ha v l e St en Sten (=5) (=2) (=2).Heney Sten (=5).Sten C=2).St en (=5) (We use almost exact computatons fo the Heney models n the above table.) Hence, we have the followng concluson hee : 87

100 Exacta bet factons > Sten (=2) > Heney > Sten (=5) > Havlle ( x > y means that we pefe x to y at % 5 level.) 4.5 Empcal analyss fo Tfecta bet Anothe type of bet n Meadowlands - Tf ecta bet means that the bettos have to guess the thee wnnng hoses n coect ode and thus, t s moe complcated than the exacta bet. Ou model fo tfecta bet wll be smla to the exacta, Le, model (4.2).Defne : tt = P(hose wns, j fnshes 2nd & k fnshes 3d}, and = P(hose k fnshes 3d J hoses S c j fnsh st S e 2nd) We have the followng logt model jk =TT T 丨 J T k J (4.4). s关 whee P 9 X A j lj = w n bet f a c t o n =P t J / (-P ), P k u = P / (-P - P ) k j a n d a) a e p a a m e t e s t o b e e s t m a t e d. Howeve, w e only have 20 aces f o t h e T f e c t a b e t In Meadowlands and hence, t h e esult h e e s not a s e l a b l e a s t h a t f o 88

101 the exacta bet n secton 4.4. Some esults ae shown n table 4.3, T a b l e 4.3 E s t m a t o n s o f T f ecta p o b a b l t e s n Meadow I ands fo 20 aces B e t t y p e / method (,,) T f e c t a b e t f a c t on 合 Wn b e t : H a v l e -7, H e n e y :ap p o x almost exact (Fo almost e x a c t computaton of Heney model, only t h e estmated pobabltes f o t h e wnnng combnatons a e computed due t o t h e l a g e amount of t m e equed n estmatng a l l z jk and thus, no paamete h a s b een estmated. In addton, w e have obtaned t f e c t a data f o t h e wmng combnatons n Hong Kong (In Hong Kong, t s called a t e c e bet). Although w e only have t h e t f e c t a b e t f a c t o n s f o t h e wnnng combnatons, w e can s t l l compute t h e log lkelhood f o t h e t f e c t a b e t f a c t o n s wthout estmatng any paametes. Some e s u l t s f o Hong Kong data (809 aces) a e shown n Table

102 Table E s t m a t o n s o f T f ecta p o b a b l t e s n Hong Kong f o 809 aces B e t t y p e / method I (,,) T f ecta bet f a c t o n W n bet ^ Have Heneyap pox almost exact ! (Fo t h e Heney model, w e have not estmated any paametes due t o t h e lage amount of memoy equed t o stoe all t h e estmated pobabltes f o 809 aces.) Fom t h e above t w o tables, by compang (,,), t appeas t h a t t h e t f ecta bet f a c t o n s the best, f o l l o w e d by t h e Heney model. The Havlle model s agan t h e wost one. Ths s fomally suppoted by t h e e s u l t s of Cox's t e s t s (smla t o e x a c t a bet compasons n t h e pevous secton) shown n Table

103 T a b l e 4.5 C o x ' s t e s t s f o t f ecta bet n Meadowlands (20 aces} T f e c t a b e t Heney ,0754 Havlle HavH e f act on T f e c t a b e t f a c t on H e n e y ( W e u s e appoxmate Heney hee.) 4.6 Empcal a n a l y s s f o t h e Qunella b e t Fo t h e Qunella bet, t h e bettos have t o guess w h c h t w o hoses f n s h f s t and second, egadless o f t h e ode and thus, t s not t h e same a s t h e exacta bet. Ths t y p e o f b e t Is n o t avalable n Meadowlands b u t t e x s t s n Hong Kong and t h e odds a e vsble t o b et t o s (on. and o f f - t a c k ). Smla t o t h e w n, e x a c t a and t f e c t a b et s, t h e qunella b e t f a c t o n should b e a choce f o estmatng t h e qunella pobablty. We d e f n e some n e w notaton f o It : J f h o s e w n s and j f n s h e s 2nd, o f h o s e j w n s and f n s h e s 2nd 0 othewse ⑶

104 and Q^= Qunella bet facton fo hoses I and j. Also, q 7 j = P ( Z = ) = tt + t j j j (I>j} Smla to the constant-p model fo the wn bet, the logt model fo the qunella bet s T J = -: Z Z0 ^ f o >j (4.6) Assumng Z - Multnomal (q7) the maxmum lkelhood estmato o f 7] can b e obtaned by maxmzng t h e f o l l o w n g log lkelhood w t h e s p e c t t o 7) z j In \ U To compae t h e qunella b e t f a c t o n w t h t h e w n bet f a c t o n usng t h e Havlle and Heney models, w e can smply eplace t h e n (4.6) b y t h e qunella pobabltes estmated b y t h e Havlle and Heney models. The above pocess eques t h e qunella b e t f a c t o n s f o a l l combnatons o f hoses and only 369 a c e s n Hong Kong a e avalable. The esults a e pesented n Table 4.6, 92

105 Table 4.6 E s t m a t o n of Q u n e l l a p o b a b l t e s f o 369 aces B e t t y p e / Model Qu n e a b e t «^ ^.. facton (l) j [(令 ^ appox almost exact ln b e t f a c t on : Havlle Heney: I f w e a e only n t e e s t e d n compang t h e accuaces among d f f e e n t b e t t y p e s and d f f e e n t models and t h e paamete estmaton s n o t consdeed, w e have many moe numbe o f a c e s (453 aces) w t h w n n n g combnatons f o makng t h e compasons and t h e e s u l t s a e s h o w n n Table 4.7. Table 4. 7 F u t h e c o m p a s o n s u s n g t h e qunella bet f o 453 aces B e t t y p e / Model 2 U n : ^ l ab facton e t ( ) Wn b e t facton : H a v H e Heney appox almost exact

106 To test fomally, we agan cay out Cox's tests and the esults ae shown n the followng table. T a b l e 4.8 C o x ' s t e s t s f o q u n e l l a bet In Hong Kong f o 369 aces Q u n e l l a b e t Heney 一2.423 facton.qunella bet Havlle facton.heney Havlle Thus, we have nconclusve esults n the fst two cases n Table 4.8. But we have a consstent esult that the Heney model s bette than the Havlle model. 4.7 Empcal analyses f o double bet and double qunella Thee ae some bet types whch nvolve moe than one ace. Double bet pays off f the two hoses chosen by a betto both wn n two consecutve aces. We have obtaned some double bet data n Meadowlands and Hong Kong. Consde Meadowlands data fst, ths contans 22 pas of aces (.e. 244 aces totally) fo double bets. We may compae the

107 accuacy of the double bet factons wth that of wn bet factons fo estmatng the pobablty of gettng a etun fo double bet. If a betto bets on hoses poft = S c j to double, the pobablty of gettng a ;J 2TT J = 7 whee and 2 ^ae the tue wnnng pobabltes of hoses and j n the fst and second aces espectvely. Defne da n ; j = t h e double b e t f a c t o n f o hoses and j n t h e f s t and second aces espectvely. Futhe, w e may u s e t h e bet 2 f a c t o n s P P t o estmate n snce t h e wnnng events n t h e t w o J s j a c e s a e ndependent othe than though possbly t h e same jockeys beng nvolved, w h e e l F^ and 2P^ a e t h e b e t f a c t o n s o f hoses and j n t h e f s t and second aces espectvely. The followng t w o expessons of l o g lkelhoods a e used f o compasons : d D o u b l e b e t f a c t o n s : T In 会 Wn bet factons : V In (P 2 P ) Y, [,+ whee the summatons ae ove all avalable pas of aces, the subscpt s used to denote the ace numbe and [] s to ndcate that the pobabltes ae assocated wth the wmng hoses. The esults ae pesented n Table 4.9.

108 T a b l e 4.9 Compason between d o u b l e bet f a c t o ns and wn bet factons n Meadowlands n 244 aces Bet types log l k D o u b l e bet f a c t o n s -452,3 Wn bet f a c t o n s -449,69 To test the hypotheses fomally. Cox's test s caed out Test Hf : double bet factons vs. H w n bet f a c t o n s g Unde H, T = f Unde H, T = g g Hence, double b e t f a c t o n s a e l e s s accuate tha n t h a t of w n bet f a c t o n s. Ths f n d n g s not supsng because bettos n t e e s t e d n double b et s have t o place t h e b e t s b e f o e t h e stat o f t h e f s t a c e and t h e y cannot obseve t h e evoluton o f odds o f double b et s. But f o w n b e t, bettos can obseve t h e evoluton of odds n both aces, whch may b e Impotant n evealng elevant n f omaton. It s nteestng t o notce t h a t t h e magnal pobablty of t h e double b e t f a c t o n can b e used a s a n estmate o f wnnng pobablty f o e a c h o f t h e t w o aces, Le. I d会;s estmates the wnnng pobablty of hose n ace, S and. 96

109 [ n estmates the wnnng pobablty of hose j In ace 2. ^fj We may compae the accuaces of these two goups of estmates wth that of wn bet factons by fttng ou constant-g models as shown n Table 4.0. Table 4. 0 Compasons o f magnal double bet factons and w n bet f a c t o n s n 244 a c e s Race Bet type l o g l k ( ) p loglko) Fst magnal w n bet magnal doube Second double w n bet (Note that the 'Fst* goup ncludes all the fst aces of the two aces f o double bets. Thus, thee ae 22 aces f o each goup.) Table 4. Compasons of magnal double bet factons and wn bet factons by Cox's tests Race H H g T f T g Fst magnal double wn bet Second magnal double wn bet

110 Fom Table 4.0, we obseve that the wn bet factons do not have a bg dffeence n ethe goup of aces (as measued by loglk(l)). Fom Table 4., Cox's tests show that the dffeence n the second goup s magnally sgnfcant (p-value fo T f = 0.054). Thus t appeas t h a t t h e d f f e e n c e o f t h e accuaces Is s l g h t l y moe s e o u s n t h e second goup o f aces. Agan, t h s s n o t supsng s n c e bettos cannot obseve t h e evoluton o f w n odds f o t h e second a c e b e f o e t h e s t a t o f t h e f s t a c e. But t h e y c a n obseve t h s n f omaton f o t h e f s t ace. In t h s analyss we may comment t h a t t h e t e s t may not b e v e y p o w e f u l s n c e only a small numbe o f a c e s a e nvolved. Now, w e t u n t o analyse Hong Kong data. The double b e t n Hong Kong s moe sophstcated and w e w l l have analyss l a t e n t h s s e c t o n. We n o w consde t h e double qunella w h c h s smply t h e combnaton o f qunella and double b e t s,. e. t h e b e t pays o f f f h o s e s and a f n s h f s t and second n t h e f s t a c e egadless o f t h e ode a n d f h o s e s j and b f n s h f s t and second n t h e second a c e egadless o f t h e ode w h e e h o s e s,a,j b a e t h e h o s e s a b e t t o h a s b e t on. We a e n t e e s t e d n compang qunella b e t f a c t o n s, double qunella b e t f a c t o n s and w n b e t f a c t o n s f o estmatng t h e t u e pobablty o f double qunella b y computng t h e e l e v a n t l o g lkelhoods. 69 p a s o f a c e s a e avalable f o double qunellas n Hong Kong. The empcal e s u l t s a e shown n 98

111 Table 4.2. Table 4.2 E s t m a t o n o f p o b a b l t e s f o t h e d o u b l e q u n e l l a bet u s n g d f f e e n t bet types n 382 aces B e t types D o u b l e q u n e l l a bet log Ilk f a c t ons Q u n e l l a bet f a c t o n s W n bet f a c t o n s : Hav e Heney (almost exact) Cox's test cannot be appled hee because we do not have the double qunella bet factons fo all combnatons- Thus we can only compae the log lkelhoods dectly. Fom Table 4.2, t can be seen that the pobabltes of double qunella deved fom the wn bet factons based on both models (Havlle and Heney) ae bette than the double qunella bet factons. Ths s consstent wth the fndng fo double bets In Meadowlands. Among these bet types, qunella bet factons ae the best and ths s s also consstent wth the fndng n the pevous secton of qunella analyss. The above analyses fo double bet and double qunella ae on the 99

112 accuacy o f pobablty estmatons. To bettos and economsts, t: may be nteestng to compae the payoffs obtaned f o double bel: and p a l a y and test whethe one Is bette than the othe. A p a l a y s not a bet type povded by the Jockey Club. It s just a way of constuctng low-pobabllty-hgh-etun bets. To make a palay smla to the double bet, we may consde bettng on hose In the fst of two consecutve aces and f t wns, bettng the ente payoff on hose j n the second ace. Asch and Quandt (987} analysed the Meadowlands data and concluded that the mean etun of the double bet s hghe than that of the palay at % 5 level of sgnfcance by usng a smple t-test. We may ty the same analyss In Hong Kong. But we ae gong to compae the double qunella bet and the palay deved fom two qunella bets and only a lttle complcaton s nvolved. To test : : Expected payoff of double qunella bet = Expected payoff of palay aganst : : Expected payoff of double qunella bet palay, o H : Expected payoff of double qunella bet > Expected payoff of palay, a smple paed compason test s employed and the t-statlstc = 3.3 (p-value=9.86x0"4),.e. we eject n favou of o at any easonable level of sgnfcance (numbe of data ponts = numbe 00

113 o f p a s of aces = 69 and thus the cental l m t theoem Is appled). To be moe e x p l c t, the above analyss s actually compang the expected values o f the f o l l o w n g two (-t ) d payoffs pe one dolla bet : (-t)2. and [2J;[2] Q Q whee t = t a c k take o f qunella bet = 0.7, t = t a c k take o f double qunella bet = 0.23, D = double qunella bet facton, fo the wnnng combnatons n the t w o aces, and 2. Qtl2» Q l2] = q u n e l l a b e t f a c t o n s f o t h e w n n n g combnatons n the two aces, espectvely. We can s e e t h a t t h e palay nvolves t w o tack ta ke s but t h e double qunella b e t nvolves one tack take only (though t h e tack tak e o f t h e double qunella b e t s hghe). It may b e moe f a, t o a c e t a n e x t e n t, t o compae t h e t w o expected p a y o f f s w h e n t h e y have equal tack take. Followng Asch and Quandt (987), w e may smply e p l a c e (-t) b y (-t ) t o equalse t h e tack takes. Then, t h e d t - s t a t s t c dops t o.3,. e. do not eject H ( p - v a l u e = 0.9), Hence, w e can say that t h e pevous sgnfcant d f f e e n c e s manly due t o t h e e f f e c t of t h e d f f e e n c e n tack takes. Ths esult s also consstent w t h Asch and Quandt's analyss f o t h e double bet on 0

114 Meadowlands data. Note that the t-test w t h the effect: o f t a c k take taken nto account s estmates equvalent to testng whethe the two pobablty (made by the qunella and the double qunella) a e sgnfcantly dffeent. The double bet n Hong Kong s moe complcated because t nvolves consolaton. The ule of payoff s as follows I f h o s e w n s t h e f s t a c e a n d h o s e j w n s t h e second a c e, t h e p a y o f f = 0.85(l-t)/D UJ f hose wns the f s t ace and hose k f n s h e s second n the second ace, the payoff = 0.5(l-t)/D (.e. I k the consolaton) whee D s the double bet facton of hoses & ;j j and t s the tack take of the double bet = 0.7. A palay whch s consdeed to be smla to ths knd of double bet s as follows If hose n t h e f s t a c e w n s, bettng t h e e n t e p a y o f f t o place o n a hose n t h e n e x t ace. Then, t pays o f f f t h a t hose comes e t h e f s t o second. In Hong Kong, w e have place b e t f o t h e a c e only when t h e numbe o f hoses s l e s s than 7. As a consequence, only 08 pas o f a c e s a e avalable f o ou analyss hee. To t e s t t h e followng hypotheses : 02

115 H E x p e c t e d p a y o f f of double b e t = E x p e c t e d p a y o f f of p a l a y, aganst : : E x p e c t e d p a y o f f of d o u b l e b e t : Expected p a y o f f o f double bet > Expected p a y o f f o f p a l a y s equvalent t o testng whethe t h e expected values o f t h e f o l l o w n g t w o t e m s a e equal (the t a c k takes o f double a n d w n bets ae equal): Double bet payoff pe one dolla bet -t l_t =0.85^ + 0.5^ [; [ 2] Palay payoff pe one dolla bet [ { [(七 E H _ ( J 2 "PI [II 2 ""PI, [2] whee D D [;] [;2] = double b e t f a c t o n o f t h e wnnng hoses n t h e t w o a c e s, = double b e t f a c t o n o f t h e wnnng ho se In t h e f s t a c e a n d t h e h o s e f n s h n g second n t h e second a c e, P 2 [] = w n b e t f a c t o n o f t h e wnnng hose In t h e f s t a c e, P = amount o f place b e t o n h o s e n t h e second ace 03

116 the subscpts [] and [2} Indcate that the bet amomnt o bet f a c t o n s assocated w t h the wnnng hose and the hose fnshes second, espectvely. The t- s ta ts tc becomes 3.93 (p-value=l.49x0 4 ) and thus H s o ejected at any easonable level of sgnfcance. Theefoe, t h s s consstent w t h the pevous f n d n g f o double qunella bet. (Note that we have no way to take nto account the tack take because of the complcated f o m of palay payoff hee.) 4.8 C o n c l u s o n s Fom a l l the above analyses, we have the f o l l o w n g conclusons : 4.8. F o the exacta bet n Meadowlands, no systematc method can be found that can be moe accuate than the exacta bet f a ct o n. S m l a l y f o the t f ecta bet n Meadowlands Agan f o the exacta bet, the magnal exacta bet factons appea to be bette than the w n bet factons f o estmatng exacta pobabltes usng any model The analyses f o double bet and double qunella bet n Meadowlands and Hong Kong show that double bet f a c t o n s and double qunella bet factons ae wose than the w n bet f a c t o n s whch, n t u n, ae not as good as the qunella bet ffactons. A smple 04

117 explanaton Is that bettos cannot obseve the f u l l evoluton of odds f o the second ace whch may ndcate some u s ef u l nfomaton at the tme they place t h e bets on the double o double qunella The expected payoffs f o double bet and double qunella bet n Hong Kong suggest that bettng on the double o double qunella bet has a hghe expected payoff than the self-constucted palays w t h the same pobabltes of gettng p o f t The Heney model s bette than the H a v l l e model (by compang t h e log lkelhoods) As the paamete fx n (4.) and (4.4) s sgnfcantly less th an one f o the Havlle model and that paamete estmate s vey close to one f o the Heney model, we conclude that the Havlle model oveestmates P(hose j fnshes 2nd wns) f j s a f a v o u t e hose and P(hose k fnshes 3 d ] wns and j fnshes 2nd) f k s a favoute hose. Ths phenomenon s known as the S l k y Sullvan. 05

118 CHAPTER FIVE DETAILED COMPARISON BETWEEN THE HARVILLE AND HENERY MODELS 5. I n t o d u c t o n I n c h a p t e f o u, smple logt models show t h a t t h e Havlle model p o d u c e s a s y s t e m a t c b a s n e s t m a t n g chapte, we wll consde estmaton of and moe In ths complcated p o b a b l t e s u s n g t h e Havlle a n d Heney models. T h e o e t c a l dscusson f o t h e d f f e e n c e b e t w e e n t h e Havlle a n d H e n e y models w l l b e ncluded hee. 5.2 The motvaton To examne moe c a e f u l l y t h e bas poduced by t h e Havlle model, w e can f t a moe complcated logt model f o P(hoses & m f n s h lst,2nd,3d,4th & 5th, espectvely) = TT = jklm ZP TT y s^ TC 丨 j P s k J S P t^ j jk j jklm as follows : m j k l Z P u^jk u j k 2 P v^i jk l v j k l (5.) (Fo example, l Ijk 06

119 fnsh st, 2nd and 3d, espectvely), etc. othe symbols ae self-explanatoy.) T h s s equvalent to the assumpton t h a t f o e a c h f n s h n g poston, the logt (n the multvaate case} of the tue c o n d t o n a l p o b a b l t y s l n e a l y e l a t e d t o t h e logt: of t h e c o n d t o n a l p o b a b l t y e s t m a t e d b y t h e H a v l l e m o d e l. Model (5.) h a s b e e n f t t e d o n a l l a v a l a b l e Hong Kong d a t a ( a c e s w t h a c e szes less t h a n 6 a e deleted f o m t h e whole d a t a s e t f o t h e c u e n t a n a l y s s ). T h e e s t m a t e d p a a m e t e s a n d t h e log lk e lh o o d b e fo e a n d a f t e e s t m a t o n a e shown n t a b l e 5.. Table 5. L o g t model f o m o e c o m p c a t e d p o b a b l t y 色 paa est. s^e , Fom t h e above analyss, w e can obseve t h a t t h e estmated p a a m e t e s a e deceasng w h e n t h e f n s h n g ode nceases. Ths n t e e s t n g obsevaton motvates u s t o study f u t h e. 07

120 5.3 C o n d t o n a l l o g s t c a n a l y s s f o t h e H a v l l e & H e n e y models To study f u t h e how good the Heney model Is when compaed to the H a v l l e model, we can f t the f o l l o w n g sees of models f o condtonal pobabltes : logt logt l o g t P to logt P 7 logt t j,. k = logt jk C logt P] logt (5.2) whee all P's on the ght hand sde of (5.2) ae the condtonal pobabltes estmated by the Havlle model and all the logts ae multvaate. Replacng all these P's by the condtonal pobabltes estmated by the Heney model, we ae able to obseve the moe geneal pattens of bas fo the two models. The log lkelhood values fo each of the above logt models wll be epoted sepaately. To smplfy ou analyss, the appoxmaton method poposed by Heney (98) s employed n ths secton snce exact computatons nvolve lots of hghe dmensonal ntegatons. Fo each ace. 08

121 P(T t <...<T l q ^ { C ^{C q T <...<T + u +u II q,q- [ S 0 > 2 0 = q- [ Z 0 q- = l n f ;n = s = l s;n q- + S 8 q- Z = s=l s;n / (n-q+) ] }. (5.3} f o q = 2 3,n-l whee C n q P ;n _ =^ ( / q P ) n q v =,, q p (pkc J F q n q = n(n-.(n-q+) = t h e x p e c t e d s t a n d a d n o m a l ode statstc,and n = total numbe of hoses n the ace. Scalng s equed to adjust the fomula n (5.3) so that all condtonal pobabltes sum to one. We have chosen hose-aces fom Hong Kong data set fo ths analyss. That s, n=8 and q=2,3,...,7. n Table 5.2 and Table The esults ae shown

122 Table 5.2 C o n d t o n a l a n a l y s s f o Hav I l e model paamete e s t mates I (paa est.) HI) Tab e 5.3 Cond tona analyss fo Heney model paamete. 358 es t mates ( ) I ( p a a est.) « (N.B. I denotes log lkelhood value and Ul) means log lkelhood value w h e n t h e appopate paamete equals one.) To obseve the systematc d f f e e n c e between t h e t w o models, t s ease t o s e e Fgues 5. and 5.2. We also show t h e d f f e e n c e of t h e log lkelhood values () between t h e t w o models n Fgue

123 0 Ju c c I n - 8 s fllk TV M n I 虞 at p s s -pa Heney mode ^, t a _ j _ 圖_園 Fgue 5-2 F g u e 5 - Estmated paametes unde Estmated paametes unde hhp. H a t v l e model the Heney model n

124 0 衾聚该 簿 Fgue 5-3 Dffeences n log lkelhood values unde the two models 2

125 Fom the above tables and fgues, t model shows a systematc bas snce the estmated paametes ae deceasng smoothly when q nceases. That means the bas poduced by the H a v l l e model s moe seous If the pobablty of a hose f n s h n g n lowe ode Is to be estmated. On the othe hand, the estmated paametes fo Heney model ae qute close to one fo dffeent q. Moeove, the log lkelhoods of the two models also show that Heney model s geneally bette especally when q s lage, but not bette than Havlle wth paametes十 5.4 Theoetcal nvestgaton of the Havlle and Heney models In ths secton, the dffeence n estmatng the condtonal pobablty of hose j fnshng second gven that hose fnshes fst by Havlle and Heney models wll be theoetcally nvestgated unde the assumpton that the Heney model s coectthat Is, we wll study the followng dffeence : t Although qute nave, the t method Is too of compang complcated to log cay lkelhood out a But It Is also of nteest to use t h s smple method (Cox( 962)5. 3 values fomal dectly test s hee.

126 "j I - TT (s 4) - Let 0.!= Expected unnng tmes of hose = E(T^. Wthout l o s s o f genealty, w e may assume : 0 < 0 < 2 <0 n (Note t h a t n 之 3. Othewse, thee s no need to dscuss n, ) 小 To study (5.4), we need the followng lemmas. Lemma 5. Let u v and w ae non-negatve functons. Moeove, u s non-deceasng and v / w s non-nceasng, then, S uv S U W f v X w O, f u s non-deceasng and v / w s non-deceasng, t h e n S uv f v 2= S U W s Poof S e e Gutmann and Maymn (987) )

127 Lemma 5.2 Defne : h(x) = whee (.) and 盃( a e standad nomal p d f and c d f, e s p e c t v e l y. Then, t c a n b e s h o w n t h a t h'(x) 2 - f o x E. Poof : h (x) _ -x$(x)0(x)-0(x)2 x ^(x) $00 [ 丽 When x^o, consde t h e functon x] the mnmum of t h s f u n c t o n occus when x=0 to show t h a t both ^(x}/^(x) and snce t s easy ( x ) / ^ ( x ) - x ae nceasng f u n c t o n s! f o xs=0. T h u s, $(x) 硕 [ $(x) 硕 _ [ t 两 ^ ( 0 ),2 _,, L 5 7 ^, > = > [ 2 競_ X Multply (5.6) by [余(x /$(x)l2, w e h a v e : h'cx) > - f o x^o. Now, f o x<0, l e t x = «y and t h u s y>0-5 ( 5-6 )

128 F s t w e c o n s d e t h a t w h e n y>0 m(y) = ^(y) m'cy) = - [ -亞(y) < 0 and w m(y) => y [ - { y ) ] > then m(y) s d e c e a s n g, = 0 b y L ' H o s p t a l ' s u l e. T h u s, m(y) > 0 f o y>0. T h e e fo e, ^ ( y ) / ^ ( - y ) N o w, d e f n e ( y ) = (l+y 2 )$(-y) l y ) = 2 [y^(-y) - ^(y)] ^ 0 and ^fy) = 0 b y (5*6.2) => (y) s non-nceasng, b y I/Hosptal's u l e. Thus, (y) ^ 0 f o y>0. N e x t, consde (5.6.3) p(y) : [l-$(y)] 2 + y0(y)[l-$(y)] It s easy t o show t h a t p'fy) = "^(y)(y) ^ 0 f o y>0, b y (5.6.3) and ⑵p(y) = 0 b y L ' H o s p t a s u l e. Theefoe p(y) (5.6.4) R e t u n n g t o ou poblem, w e consde, f o x 0, ^ ( X },2 JUT $(X) 0(x) J -My) <PCy), -^Cy) (7) = p(y) > Q (5.7) 沴(y 2 f o y>0 b y (5.6.4). F n a l l y, f o m (5.6) a n d (5.7), w e h a v e : h ' C x ) 之 - f o x IR. L e m m a 5.3: Defne t h e followng functon J(v: 0 ) (>(v+0-0 ) T I ^ ( v ) Y J ^j [<^(v+0 - )尤^ ^(v-0^+0^) s s s 关 6 (5.8)

129 then, JCv; 9 ) Is non-deceasng In v, and J(v; 0 s n o n - n c e a s n g n v, O whee a = M n,.e. 0 = Mn 0 a 共 *l and b = Max, I.e. 0 = Max 0 Poof : C o n s d e t h e devatve o f J(v 0^) n (5.8) w t h e s p e c t t o d J(v; 0 ) J 3 C I V 2 { 5 > ( V : s s^l j V+0 0 ^ ) ) J $ ( v t 萁 s ) > t ) <p(v-q +0 ) J] 0 ( V+0 0^) S s^j E s^l S $ ( V 0 ^ +0^) y _L_ 士 $(v-e +0 ) ^ J (v+0-0 ] 沴(v 0 +0 ) 0 (v+0 0 ) T I $ (y 0 +0 ) [ y ---c- (v+8 ^ s t^s t ^ $ ( y - 0 +e ) s t^s 0) ] 0Y ( v ) $ (v~0 +0 ) j ^j I 0(v+0 0 j 0^+0^} ^(y 0 +0 ) L $ ( v ) J 少 v 0 +0 ) y 3 t ^ ( v ) t^s t

130 V+0^-0^) IJ^$(v-0^+0 } V-0 +0 <p(v-q + 0 $ (V 0 +0 ) $(v +0 (0. whee C = Y^(v+0-0 ) T I ^(v-0 +0 ) s s t^s t Recall t h e d e f n t o n o f h(x) n Lemma 5.2, then, f j=a. d J(v: s^l a 0 ( v + 0-0^) ^ C $(v 0^0^) h(v-0 +0 ) - h(v-0 +0 ) + (0-0 ) (9^0) ^(V+0a"0l)a$(V'"0I+0) [hf{v ) + ] ( 0-9 ) S^l s n c e h(v-0+ 0 ) - h(v ) = h'tv ) (8-0 ) s a 0 s a b y t h e mean value theoem, whee v (v-0 +0 v-0+0 ) by Lemma 5.2 8

131 On t h e othe hand, f j=b d J(v; 0 ) b { Z 乂 =<ACv+0 -e ) n b *lb 树 ^ ) 具 广 } $(v-0 +9 Y * C h ( v 0 +0 ) h(v-0 +0 ) (0 8 ) s b b s U^b s L J \ (0 =沴 v+g - e ) n $ ( v ) y b ^b ( I ^ b s c [ [ - h l v ) - Ce - e ) Y 0 b s J- J s n c e h(v-0 +0 ) - h(v ) = - h ' t v ) (0 一 s I b by the mean value theoem, whee v (v-0 +0, v ) 0 s, ^ 0 b y Lemma 5. 2 We have s h o w n t h a t d J(v; 0 ) a d J(v 8^) b hence t h e esult follows. 9 b O b

132 Theoem 5. n a ' - b TT I ' - T C whee a = M n. e. 0 = Mn 0 and Max,.e. 8 = Max 0 b ^ ^ Poof Consde t h e d f f e e n c e (5.4), V - n n j [ $(u 0 +0 )] j I s^j TI $( 0 +0 ) 0(u) H $(u-0 +0 ) s^ij J s j s J 内 J whee n / = P(T < Mn {T }) s j⑴ J. s^l J Le. the pobablty of hose j wns f hose s emoved fom the ace. Theefoe. 20

133 D e f n e gs.i =ss z, zj ja) (5.9) Thus, t s u f f c e s t o show that g,a ^ 0 and g b, : l l Now, w e c o n s d e : ⑴ 3 工 s^j J t -00 $ ( u - 0 +e ) 0(u) d u j l s $ t u " 0 s + 0 t 沴 ( u d S u

134 IJ ^(u-0 T I $(-e y t t5s金 ( U " * 0 s + e t ) 沴 u ) d u -oo E s关 j ^ S^ j + 0 ) +0 )沴(u) du t^s -00 s $(U-0 j t + 0 ) K (-0 +0 )Y () d u ^j j $ ( u ) T I $ ( u» 0 +0 ) ^ ( u ) d u s t^s s t s^ j g l $(v-0^ +0 ) 々 ( v - Q + Q d v ^ (v-g ^ +0 ^) (j) (v 0 ^ +0 ) dv s^ $(v) $(V) o H J ^(v 0 u t式s s忒 j +0 ) ^(v-0 +0 ) dv t j s by change of vaables usng : v = u-0 +0 n the numeato, and j v = u-0 +0 n the denomnato 22

135 0 when j=a \ - b y usng Lemma 5.3 togethe wth Lemma w h e n j=b Hence 0 and 2 0 and t h e e q u e d e s u l t f o l l o w s, 匾 When n>3, we have only shown that the above esult s vald fo exteme values of j. But fo a<j<b, the dffeence may be geate than o smalle than zeo dependng on the patcula set of (0,0,...,0 ) 2 n The above t h e o e m means t h a t f t h e unnng t m e s s a t s f y t h e assumpton o f t h e Heney model, t h e Havlle model w l l oveestmate t h e condtonal pobablty o f t h e most f a v o u t e h o s e f n s h n g second and undeestmate t h e condtonal pobablty o f t h e longshot f n s h n g second. 5.5 C o n c l u s o n By t h e e s u l t s obtaned n t h s chapte, based on ou Hong Kong d a t a s e t, w e f u t h e suppot t h e concluson t h a t t h e Havlle model h a s s y s t e m a t c b a s n estmatng moe complcated pobabltes. A dscountng p a t t e n o f estmated paametes s a l s o suggested n 23

136 sectons 5.2 and 5.3. On the othe hand, t h e Heney model does not cause any systematc bas and thus It s h o u l d b e m o e e l a b l e. R e c a l l t h a t H a u s c h a n d Z e m b a (98) u s e d t h e H a v l l e mode l f o estmatng exacta and t fe c t a pobabltes and put all the e s t m a t e d p o b a b l t e s n t o t h e D. Z s y s t e m f o t h e p u p o s e of m a x m z a t o n o f a t e o f e t u n. Hence, w e e x p e c t t h e s y s t e m w l l b e m o e e l a b l e f t h e y u s e t h e Heney model nstead. Howeve, t h s does n o t necessaly mean that t h e p o f t s obtaned w l l b e g e a t e. We w l l e t u n t o t h s b e t t n g s t a t e g y n C h a p t e e g h t. 24

137 CHAPTER SIX EXTENSIONS OF THE HENERY MODEL 6. Intoducton By usng the esults obtaned n Chaptes fou and fve, based on patcula data sets we may conclude t h a t t h e Heney model s q u t e e l a b l e w h e n compaed t o t h e Havlle m o d e U n t h s Chapte, w e e x t e n d Heney's Idea t o nonconstant vaance and t h e dependent unnng t m e s case. The objectve s t o e s t m a t e t h e complcated pobabltes n a bette way. 6.2 The s e l e c t e d model R e c a l l t h a t Heney (98) assumed t h e unnng t m e s o f hoses : T - N(0,) ndependently. Two questons a e a s e d : ⑴ Hs method s equvalent to a constant vaance model but s the constant vaance assumpton always vald () T, " T m a y n o t b e t u l y ndependent. Intutvely, w e thnk n t h a t t h e unnng t m e s o f hoses should b e dependent. In patcula, stonge h o s e s may have hghe coelaton o f unnng t m e s t h a n t h e w e a k e hoses. Fo example, f one stong ho se u n s f a s t e t h a n t s mean t m e, anothe equally stong hose w l l t a c e t b y unnng 25

138 faste, too. In fact, f any monotona nceasng tansf om of s dstbuted as NCQ^l), the Heney model emans unchanged and ths fact s obvous by consdeng the odeng pobablty. In patcula, the log nomal unnng tmes model wll be equvalent to the Heney model. A smple extenson s the constant coelaton model as follows T N n (0,E), 一 whee 0 = ( 0 一.e.,..., 0 )nt a n d E = c2 (6.) coct.,t^) = p V I, j then, t can be shown that : Theoem 6. 丁 0 c =Vp U o + vt^p U Id whee U U,...,U o n N(0 (Fo t h e poof o f t h s theoem, s e e Tong Cl990»p.2-3).) 26 (6.2)

139 Theoem 6.2 The estmaton pocedue of odeng pobabltes unde the Heney model emans the same when the coelaton of unnng tmes s a constant, p. Poof Unde (6.) we have : P T <...<T ) = P(T / n t <... < T / c) n =PCB^/ c + Vl-p U < < G / c + vt=p U ) n n by (6.2) =l,2, n Now d e f n e V whee cvt n d e p t s c e a t h a t V ^ ld - N ( 0,) snc e U - N(0,) T h e e f o e, P(T <...<T ) ovl-p ovt = P ( V <...<v ) n Hence, t h e Heney model s equvalent t o a constant coelaton model f o estmatng complcated pobabltes, I.e. t h e pocedue t o 27

140 obtan an estmate of _0 (now 0 ), etc. Is the same.画 醒 mmm Theefoe, to mpove the Heney model, we should look fo a moe sophstcated nodel Two Ideas ae as folows 6.2. N o n - c o n s t a n t c o e l a t o n c a s e We would lke t o f n d a model so that Heney's Idea of Taylo s e e s expanson s t l l woks. Othewse, t would not b e feasble t o evaluate so many multvaate Integals. A common non-constant coelaton model s = V,j (6.3) t h a t s, t h e coelaton s sepaable nto t w o pats. It s called stuctue- by some eseaches (e.g. Tong(980)). Smla t o theoem 6., w e now have t h e followng theoem. Theoem 6.3 Unde t h e patcula coelaton stuctue (6.3), w e have : T 0 j ^ = lf V Q + / whee U =,2,..., n (6.4) d U,U,..., - N(0,) o n. e. t h e d e p e n d e n c e of T^ s due t o t h e andom vaable (see Johnson and Kotz (972, p.47} f o detals) 28 only.

141 F o t h s p a t c u l a coelaton stuctue, we have the f o l l o w n g s m p l f c a t o n f o computng odeng pobabltes : P C X <...<T ) = f n 0 J F ( T < ". < T I u ) ^ ( u )d u n o 0 0 P(V <...<V S u )沴(u ) du n 0 0 (6.5) 0 b y (6.4) whee V = + Indep t s c l e a t h a t V l u - N ( ~ u, -0 2 ) 0 o l o.e. V,V 2 (o n T»T,...,T ) 2 n (6.6) ae condtonally ndependent gven T h e e fo e, u n d e t h e model 6.3), t h e o d e n g p o b a b l t y n v o l v e s o n l y o n e m o e n t e g a l. As s e e n n (6.5), t h e p o b a b l t y n s d e t h e n t e g a l s t h e s a m e a s t h e o g n a l f o m of o d e n g p o b a b l t y f o t h e H e n e y model. To m o d e l t h e e l a t o n s h p b e t w e e n t h e c o e l a t o n a n d m e a n u n n n g tmes, w e popose : I n (-=~j ) = f - d (0-0) o, whee l + exp(f+<(0^- 0) 0 = (S 0 ) / n 29 (6.7)

142 By consdeng a pa o f hoses unnng n the acetack, f both hoses ae stong (.e. t h e e^'s ae e l a t v e l y small), we expect the coelaton w l l be hghe and vce vesa. That means, t h e coelaton o f unnng "tmes between two hoses depends on the mean unnng tmes o f both hoses. When consdeng as a functon of (0^- 0), f f=0, the pont o f n f l e x o n Is at ( 0 ^ 0) = 0. Othewse, t w l l be a t - f / d. Thus, f contols the poston of p o n t of nflexon and d contols the dependence o f the coelaton on mean unnng tmes. Note that ths model f o c e s p to l e between 0 and, and thus p j w l l also l e between 0 and. Unde ths patcula model, T j (We assume ndep u o - N(0广 -^) by (6.6) 2 < = wthout loss of genealty.) 6,2.2 Non-constant vaance case Apat f o m non-constant coelaton, we can have a non-constant vaance model. The poposed f o m of vaance s : c^ = explbfo^- 0)] (6.8) That means, f b>0, weake hoses w l l have hghe vaance o f unnng tmes and stonge hoses w l l have moe stable unnng tmes and ths s qute sensble. In patcula, f b:0, (6.8) educes to the Heney model. 30

143 6.3 Appoxmaton f o m u l a s I n o d e t o o b t a n t h e M L E s f o "the p a a m e t e s b c a n d f, w e choose t h e t o p f v e postons f o constuctng t h e lkelhood functon because t h e coelaton and non-constant vaance stuctues should have moe e f f e c t o n estmatng moe complcated pobabltes. Howeve, t h o s e pobabltes a e cetanly too complcated t o b e computed and thus some appoxmaton methods should b e moe appopate. Usng Heney,s dea o f a Taylo expanson, t can b e shown that <pvz ) ( n - l ) C z 一 z 0,, 一 ~ ^ whee z = $ ( l / n ), ) (6.9) =念卞 = w n bet f a c t o n f o hose I, M =A,. 严 ; n ^;n u = t h expected standad nomal ode statstc, l;n (2) = th second moment about ogn of standad ;n nomal ode statstc. 3

144 let f 9 = +e and =l-l/n s then U L c V 漏b f f 2-) V - l/f 9» and T h e d e v a t o n o f ( 6, 9 ) s s h o w n n A p p e n d x C. Moeove, P(T CT mn 5 $ { C ^, 5 5 {T > ) = + [ Z 0 M 5,2,3» 4-, 5 +S 0 = y $ { C + v [ E 0 M + E 0 5,2,3, t4,t5 whee C 5 = l t S M / (n-5) > =l s = l =l t 2 M / (n-5)] > s=l C6.0) = $ (/ P ) 5 0(C5) n ( n - l )...(n~5+l) T h e t e d o u s d e v a t o n o f (6.0) s a l s o g v e n n A p p e n d x C. H e n c e, t h e l o g l k e l h o o d s : m = I! I n tt 乙 = whee n [2345,, (6,) 2345 ], s the pobablty of the top fve hoses actually fnshng ace n the coect ode. Maxmzng (6.) wth espect 32

145 t o appopate paametes s staghtfowad the though the fom of s e c o n d d e v a t v e s o f (6.} s n o t so s m p l e, 6.4 E m p c a l e s u l t s Recall the two extended vesons of the Heney model ae : ⑴ () p j ^ w h e e = { + e x p [ f + d ( 0 exp[b(0 ^ - 0)] w h e e f, d and b a e paametes t o b e estmated. T h e a b o v e m o d e l s h a v e b e e n f t t e d o n h o s e - a c e s n Hong Ko n g. T h e m o d e l s t o p e d c t t h e p o b a b l t e s o f h o s e s f n s h n g n t h e t o p f v e postons. The empcal esults a e summazed n T a b l e

146 Table 6. Compason between Heney model and extended models E s t m a t es loglik p~va ue Heney model Co s t uctue (d only ) Co s t u c t u e » (d,f) V a st u c t u e (b only} (Note In t h s t a b l e, p - v a l u e n d c a t e s t h e s g n f c a n c e o f t h e d f f e e n c e b e t w e e n t h e e x t e n d e d model and t h e ognal Heney model b y t h e lkelhood ato test.) H e n c e, t h e a d d t o n o f v a a n c e ( p a a m e t e b) o c o e l a t o n, ( p a a m e t e d) s t u c t u e a p p e a s t o b e n s g n f c a n t a t 5Z l e v e l b u t s g n f c a n t a t 0% l e v e l. To v e f y t h a t t h e a p p o x m a t o n f o m u l a s n o t t o o bad w t h e s p e c t t o e s t m a t o n o f t h e p a a m e t e s, s m u l a t o n s a e used and t h e esults ae epoted n t h e followng table. 34

147 T ab e 6.2 S m u l a t o n e s u l t s f o t h e e x t e n d e d Heney model f ) 08 ) oo 0.05 paamete estmates logl k (tue) d = b = d = b = 0, d logl k ( p a a, est:).0798 In the above table, the same data set as f o the pevous table (Table 6.) s used but the f n s h n g ode s smulated w t h ethe a coelaton stuctue o a vaance stuctue f o each ace. Thus ths smulaton execse vefes that the appoxmaton fomula f o the estmaton of paametes s not too bad. 35

148 CHAPTER SEVEN A RECOMMENDED SIMPLE MODEL 7. I n t o d u c t o n I n Chapte f o u, we have compaed the Havlle and Heney models usng logt models- The log lkelhood f o the H a v l l e model a f t e paamete estmaton, heeafte called the dscount model due to the dscountng patten obseved n the analyses mentoned n Chapte f v e, s close to the log lkelhood f o the Heney model. In t h s chapte, we w l l dscuss ths obsevaton. Moeove, a smple model s ecommended to appoxmate the Heney model. 7.2 S m l a t y between t h e t w o models F o the Heney model, t nvolves ntegatons and j whee P =wn bet f a c t o n P, = P / (-P ) 36

149 Theefoe, w e have A 广 S s - 咖 Usng ou Meadowlands data, we compae the model (7.) w t h the Heney model by dectly obsevng t h e log lkelhood values and the esult o f Cox's test Let Heney H H a v l l e p l u s j only % loglk f o = , loglk f o H g = , Cox ' s t e s t e s u l t T = f T = g T h e e f o e, n o s g n f c a n t d f f e e n c e e x s t s b e t w e e n t h e t w o models. Now, suppose ^ and a e t h e wnnng pobablty and exacta p o b a b l t y e s t m a t e d b y t h e H e n e y model, e s p e c t v e l y. 37 s b a T - t o h s E n o e y u.¾ o V a f e h t t u o b a o w 0 t o n e d w f e IX. p m s h c u m s f o w n b e t f a c t o n s, w e may s e t = and assume

150 Defne.Hen,,,= j In ft / n ) Ij II (7.2) ln(7 / t ) We c a n co mp u te t h s t e m f o a n y combnaton a c e s. T h sx : of, j, l n a l l t h e s v e y close t o a c o n s t a n t f o d f f e e n t j l a n d n d f f e e n t a c e s. Thus, b a s e d on (7.2), can b e estmated d e c t l y n s t e a d of u s n g n u m e c a l n t e g a t o n s. We w l l e v s t t h s p o n t n t h e n e x t secton. I t c a n b e f o u n d I n t h e H.K. a n d Meadowlands d a t a s e t s t h a t t h e X H e n Is close t o X H e n = 0.76 w h c h s t s e l f close t o t h e v a l u e of j, o b t a n e d b y MLE u s n g t h e f o l l o w n g models : l o g t tt, = fx J l Hav o g t n, = n, a j (wn b e t ) (7.3) l o g t t, H e n 小 logt P 小 (7.4) a whee tt 丨=tt j U and T ae obtaned by Heney,s appoxmaton usng w n bets. J 7 E m p c a l esults f o the models I n (7.3) and C7.4) have aleady been dscussed n Chapte fouthe eason w h y AHcn «A H a v / ^ H e n 38 s a s follows :

151 I f (7.3) and ( 7. 4 ) a e tue ^Hav 7 ^Hen = l 0 g t 会j 卜 ^ = 0 g t < = 諫 小 / P 小 ^ I n ( / tt ) 八, 八 ln(7j / InfP ; f o any / xhen j J TI ) T h e e f oe, a n estmato o f A Hen s gven b yj & that, e m p c a l l y, j between 八 严Hav ^Hen Is qute close t o one. Thus, the d f f e e n c e Hen and 入Hen s qute small. T h s s the eason w h y the dscount model based on u Hen and the Heney model have s m l a pecson. We can also eplace the w n bet f a c t o n s n models (7.3) and (7.4) b y m a g n a l exacta bet f a c t o n s a n d the a t o o f t h e MLEs s also v e y close t o the pevous a t o o f t h e ML Es w h c h Moeove, w e obseve t h a t t h e 入二 Is s l g h t l y dependent on d e c t l y. Some summay values o f a e shown n Table Hen n f

152 Tab le 7. S u m m a y v a l u e s o f A. H e n J Racet ack mean Hong K o n g ( 8 9 ) standad devaton Meadowlands Japan (In t h e above table,, j a e hoses f n s h n g f s t and second, e s p e c t v e l y. Hose (萁,j) vaes.) F u t h e, s m l a obsevaton can be f o u n d f o, w h e e t h=e n ln(7t /tt ) ( l j k l ln(^ k 7. 5 ) ) Agan, f o m o u d a t a, w e o b s e v e t h a t t h s t h e n s e s t m a t e d t o b e b y u s n g a a t o of MLE o f t w o p a a m e t e s s m l a t o t h e above. T h e s u m m a y v a l u e s of a e shown n Table 7.2. Table 7.2 Summay values of Racetack mean standad dev aton H o n g K o n g (89) , Meadowlands Japan , (In t h e a b o v e t a b l e, j a n d k a e h o s e s f n s h n g f s t, second a n d t h d, e s p e c t v e l y. H o s e (萁,j,k) vaes.) 40

153 Although the mean o f Is close to 0.64 o 0.65 In. the above table, usng 0.62 does not make a b g d f f e e n c e. Ths can be seen by compang the use o f 0.62 and 0.65 as shown In Table 7.3. I n t h s t a b l e, 入 H e n s f x e d at F o each selected T H e n, we f t a constant-p model. We can see t h a t t h e e Is not a b g d f f e e n c e n the log l k e l h o o d values when t Hen s set t o 0.62 and Table 7.3 E m p c a l c o m p a s o n s u s n g x Race t a c k s Hong K o n g Meadowlands THen Hen = & p () I (g) To undestand the e f f e c t o f a c e s z e on A Hen and *Hen, we can compute the summay values f o d f f e e n t ace szes. Ths s shown n t h e f o l l o w n g tables. 4

154 Table A (a) Summay values o f 入Hen and THen f o d f f e e n t ace szes n Hong Kong (89) ace sze no.of aces mean s. d. mean , T a b l e 7. 4 (b) S u m m a y v a l u e s o f XHen and T Hen f o d f f e e n t a c e s z e s n Meadowlands ace s z e n o. o f aces mean s. d. mean

155 Table 7.4 (c) Summay values of AHen and THen f o d f f e e n t ace szes n Japan ace s z e no.of aces mean X Hen Hen mean 6, F o m t h e above t a b l e s, w e s e e t h a t XHen a n d T Hen have a n n c e a s n g t e n d a s t h e a c e s z e n n c e a s e s b u t t h e values do n o t v a y a l o t. I n f a c t w e may s u m m a z e t h e e l a t o n s h p s usng t h e f o l l o w n g a d - h o c eg es s on models based on t h e m e a n v a l u e s shown I n T a b l e 7.4 ( a ) : AHen = n n 2 R 2 = 98.7 o o t mse =

156 T H e n = n n 2 R 2 = 99.0 oot mse = o, usng logt tansfomatons on the LHS, A Hen logt A R 2 = log n = o, oot mse = logt T Hen = log n R 2 = 98.2 % oot m s e = W t h f x e d v a l u e s of 入Hen= 0.76 and THen= 0.62, t h e log lkelhood value usng the data used n Table 7.4 (a) f o t h e f s t thee f n s h n g o d e egessons, t h e log lkelhood values a e and -2502,55 espectvely. Hence, the e f f e c t of the dependence o f the log l k e l h oo d on a c e s z e s small. 7.3 Some s m u l a t o n e s u l t s In the above secton, we have epoted t h e summay statstcs o f 入二 based on the avalable data sets. In t h s secton, w e w l l pesent some smulaton esults t o suppot ou Idea o f {c( H d Assume t h a t, f o e a c h a c e, s ^^ 0, c 0 ^ 丨0% =,2,.-.,, whee a n a b t a y c o n s t a n t (snce o d e n g p o b a b l t e s depend on 2 t h e d f f e e n c e b e t w e e n 0 ' s only) a n d c s a p e s p e c f e d value* T h e o v a l u e c can b e n t e p e t e d a s a m e a s u e of d s p e s o n o f t h e m e a n u n n n g t m e s of t h e h o s e s n t h e s a m e a c e. I n o t h e w o d s, t 44

157 measues the v a a t o n o f abltes (o wnnng pobabltes) o f the hoses n the a c e. Based on t h s assumpton, we can use Monte Calo then compute A Hen usng (7.2). s m u l a t o n o f 0 =( 0 ^, a n d To detemne a sutable n (the numbe o f hoses n a ace) f o t h e smulaton, w e compute the means and standad devatons o f the numbe o f hoses f o the acetacks t h a t we ae most nteested n (see Table 7.5). Table Mean a n d s t a n d a d d e v a t o n o f n f o e a c h a c e t a c k Racet ack Mean s.d Meadowlands ( U. S. ) Japan Hong K o n g ( ) A s t h e means a e qute close to ten, w e have set n=0. We t y Hen d f f e e n t c t o obseve t s nfluence on t h e A. F f t y aces a e 0 ]l smulated f o each T ( Fo each a c e, w e have f x e d t w o hoses f o o & j b u t s vayng ove t h e o t h e hoses a n d thus, w e have eght A H e n s f o each ace. Theefoe, t h e e ae 50x8 = 400 j U each smulaton. The smulaton es ults a e shown n Table

158 Table 7.6 S m u l a t o n s o f X H e n f o n=0 a n d 5 0 aces mean o f of F o m Table 7.6, we obseve that the mean value o f Hen to 0,76 f o a l l c values t h o u g h t h e s t a n d a d d e v a t o n d e p e n d s on T h s m e a n s t h e t h e a c c u a c y of o u f x e d A model d e p e n d s o n c^. L a g e c may a f f e c t o u a p p o x m a t o n of tc.^ S m u l a t o n e s u l t s f o o t h e e x t e m e v a l u e s of n a e s h o w n n T a b l e 7,7 a n d 7.8. Though t h e m e a n v a l u e s d e v a t e f o m 0.76 a l t t l e, t h e d f f e e n c e s a e q u t e s m a l l a n d t h u s u s n g 0.76 should n o t h a v e a n y s e o u s a d v e s e e f f e c t o n t h e e s t m a t o n of n. j Tab e 7. 7 S m u l a t o n s o f 入 H e n f o n=7 a n d 5 0 a c e s jk mean o f Hen d. o f X

159 Table 7.8 Hen Smulatons o f X f o n=4 a n d 50 a c e s mean o f A I n e a l t y, the of X should not be too lage snce t h e jockey club o f each a c e t a c k w l l t y equalzng the abltes of hoses n t h e same ace. F o nstance, the hoses ae usually n the same class w h c h ndcates the a b l t y level o f the hoses. A l s o t h e j o c k e y c l u b c a n p u t d f f e e n t w e g h t s on t h e h o s e s so a s t o e d u c e t h e d e v a t o n of t h e w n n n g c h a n c e s. F o t h e e a l d a t a, w e assume N(^,c^) n d e p e n d e n t l y w h e e j denotes t h e a c e numbe and denotes t h e hose numbe n a c e j. I n f a c t, each s e s t m a t e d b y H e n e y ' s a p p o x m a t o n m e t h o d (see 2 C h a p t e f o u f o d e t a l s ). T h e n a n u n b a s e d e s t m a t o f o s g v e n (7.6) I (n-) 47

160 Z ( e 々 whee If t h e v a a n c e of 0 s n o t a c o n s t a n t, j ndep Le. 2 NCju^c^), w e can compute t h e sample s t a n d a d devaton c f o e a c h a c e and t h e s u mma y s t a t s t c s f o c^ a e shown n Table 7.9. T a b l e.9 八 S u m m a y s t a t s t c s f o c^ and t h e B a t l e t t s t a t s t c s " R a c e t a c k #aces m e a n o f c j j statstc ~ Hong K o n g (85-89 ) Meadowlands Japan c^ a e q u t e small w h e n compaed t o t h e means. T h e B a t l e t t t e s t (e.g. s e e Montgomey (99)) s employed t o t e s t t h e e q u a l t y of v a a n c e s n d f f e e n t a c e s. T h e B a t l e t t s t a t s t c s a e a l s o shown n t h e above t a b l e t o g e t h e w t h t h e c t c a l values f o compason. T h e c t c a l v a l u e s a e x 2 {v) w t h 57 sgnfcance level a n d t h e d e g e e o f f e e d o m * v = of aces Fom Table 7.9» t h e s t a n d a d d e v a t o n s of numbe value - L These ^values a e obtaned f o m a n a p p o x m a t o n f o m u l a f o l a g e d e g e e s of f e e d o m n Chu (978). 48

161 Theefoe, we can obseve that the dffeence n vaances Is not: s g n f c a n t. In f a c t, the Batlett statstcs ae a l l less than the 2 means o f x { v ) w h c h ae just the degees of feedom (numbe of aces - ) and thus, sgnfcance they ae not level. Hence, we standad devaton of sgnfcant poceed to at any easonable estmate the oveall f o each acetack by usng (7.6)- The e s u l t s ae shown n Table 7.0. T a b l e 7.0 Estmaton of c f o each a c e t a c k o Racetack # aces A c o Hong Kong (85-89) Meadowlands Japan Thus, f o m Table 7.0, the estmated standad devaton of mean unnng tmes s qute small and ths obsevaton matches wth the nce smulaton esults wth small devatons o f X Hen ae expected). 49 c o (and thus small standad

162 7.4 E m p c a l a n a l y s s u s n g f x e d l a m b d a a n d t a u I n Chapte f o u w e h a v e c o m p a e d d f f e e n t models f o estmatng exacta, qunella and Ixfecta pobabltes. In ths secton we w l l u s e f x e d X a n d x t o c o m p a e w t h p e v o u s models. Some e m p c a l e s u l t s a e s h o w n n T a b l e 7. a n d T a b l e 7-2. Log l k e l h o o d s of s o m e p e v o u s models a e a l s o s h o w n f o e f e e n c e s. Tab e 7. Compasons among dffeent models ModeI s E x a c t a : 50 a c e s ( M e a d o w I a n d s ) fxed X Havle Hav lle+logt Heney T f e c t a : 20 a c e s ( M e a d o w l a n d s ) f x e d 入 t: H a v l e Hav lle+logt Heney T f e c t a : a c e s ( H on g K o n g ) f x e d X, t Havlle H a v le+log t Heney

163 Note : () A l l the above esults ae obtaned by usng w n be: factons. () Havlle+logt = dscount model. () F o exacta and t f e c t a, f x e d X Hen and t h e n mean that \ H e n eplaces paamete p should be estmated f o adjustng the favoute-longshot bas. (v) A l l log lkelhood values f o Havlle+logt's ae assocated w t h the MLEs. (v) F o the Heney models, a l l the esults shown hee ae obtaned by almost exact computatons. T a b l e 7. 2 Othe compasons w t h o u t paamete estmaton u s n g Hong Kong d a t a M o de l s log l k e l h o o d Q u n e l a : ( a c e s ) fxed 入 Havlle Heney Double qunella : (69 p a s o f a c e s ) fxed A Havlle Heney (The above table does not gve estmates o f (3 f o favoutte-longshot bas because t s not feasble to estmaton o f qunella b e t j 5 do so f o the pobablty

164 F o m Table 7. and Table 7.2» w e o b s e v e t h a t t h e a c c u a c y ( m e a s u e d b y t h e log lkelhoods) of t h e f x e d A a n d model s c l o s e t o t h a t of t h e H e n e y m o d e l a n d t h e H a v l l e model w t h m p o v e m e n t m a d e b y l o g t m o de ls, 7.5 M a t c h n g b e t w e e n t h e H e n e y m o d e l a n d E x a c t a foel: f a c t o n s I n t h s s e c t o n, s m l a t o X Hen, w e d e f n e = jl ln(ep n / E P u n p P, w h e e E P^ s t h e e x a c t a b e t f a c t o n ( n Meadowlands), s the magnal exacta, and t h e supescpt Xz d e n o t e s g a m b l n g,. e. t h s t e m g of coesponds t o t h e a t o of logts poduced b y actual gambles (exacta b e t f a c t o n s hee)- We a s s u m e t h a t E P U s a b e t t e e s t m a t e of t h e e x a c t a p o b a b l t y t h a n s t h e p o b a b l t y e s t m a t e d b y a l l t h e o t h e models dscussed so f a. Although no f m vefcaton s povded, empcal e s u l t s obtaned n Chapte f o u suggest t h a t EP^ s t h e b e s t. If Heney's nomal assumpton o f u n n n g tmes s t u e, w e wll e x p e c t t h a t X Hen a n d w l l b e q u t e close t o e a c h o t h e o a t l e a s t h a v e s m l a p a t t e n. Howeve, t h s s n o t s o a n d n f a c t, w e obseve 52

165 that t h s 入^ s cetanly not close to a constant: and t s vaaton s supsngly lage. Also, the gaphcal elatonshp between 入 and 入: s not clea and t h e coelatons ae vey weak though j they ae sgnfcant. The gaph and the coelatons ae shown n Fgue 7. and Table 7.3, espectvely. T a b l e 7.3 C o e l a t o n s b e t w e e n A H e n a n d Xz Type coelaton Peas on Speaman Kenda ' s t a u

166 F g u e 7. lombda-g 瓜 lambda-tm* 0,74 0,76 amfcda-h«n

167 Hence, unde the assumpton that the exacta bet factons ae bette estmates o f exacta pobabltes, we do suppot that Heney model s bette than the f x e d - A snce入 ⑶ not close to Hen s j 7.6 O t h e a p p o x m a t o n s f o t h e H e n e y m o d e l R e c a l l t h a t t h e u s e of t h e H e n e y model nvolves t w o s t a g e s : () Compute 0 b a s e d o n t h e b e t f a c t o n s P () Compute moe complcated pobabltes based on the 0 obtaned n (). The appoxmatons used n both stages suggested n Heney (98) ae smply f s t - o d e Taylo sees appoxmatons. Hence, anothe s t a g h t f o w a d dea to mpove the appoxmatons s to use a second-ode T a y l o sees expanson. It s vey tedous to deve the appoxmaton fomulas n the two stages, The detals of devaton o f the appoxmatons f o both stage ae pesented n Appendx D. v + 沴(z ) ( n - l ) o I 0 士 A 2 f o =, J A 02 + B E B 0 0 ) 2 j^k J k k^i J, 55 () and (II) above

168 l, whee (p^zy (2) n (7. J + 2 n(n-l) Cn-) (2) ^;n (pxy - 2 ( ^( z) <f>^ y (2), n(n-l)(n-2) n(n-), (2) (puy n(n-l) (2: J 一 n(n-l) 5 t h e f s t a n d second moments (about z e o ) of nlmnum s t a n d a d n o m a l o d e s t a t s t c of s z e n. (7.7) s a s y s t e m of q u a d a t c e q u a t o n s a n d t h u s n u m e c a l m e t h o d s a e e q u e d t o solve t. To mp o v e t h e ap p o x m a t o n n s t a g e () u s n g a s e c o n d - o d e T a y l o s e e s, w e h av e : 56

169 (7,8) whee n(n-) n(n-l)0(a) $ - ( 沴 a)' I 0=0 [^(a) 0J2 [沴(a) I 3 + a I 4 2 ] + I k^j k [炎(a) L + al / E E % l^k 5 6 d Qd Q 0=0 57

170 2 E 0k [ _ k^i j 2 0 E k^j 8 + a I 6 I2 Ie t如a) j [ (3) l + ai 4 I 2 ] (2 ) 2 n ( n - l ) ' n(n-l)' n(n-) (2) (2 ) ncn~l)(n-2) n(n-)cn-2) 0 o n (-) (2) - 2 (3-2/ (2). n 2 (-j J 2 n 8 n(n-l)cn-2)(n-3) ( 2 ), n(n-l)(n-2) To compae t h e above appoxmaton w t h t h e othes, w e can compute t h e log lkelhoods and t h e e s u l t s a e summazed n the followng table. 58

171 Table 7.4 C o m p a s o n s a m o n g d f f e e n t m e t h o d s f o t h e Heney model b a s e d o n e x a c t a b e t n Meadowlands (50 a c e s ) Mode s log l k () s t od e T a y l o sees f o a) s t o d e f o tt b) 2 n d od e f o J j c ) N u m e c a l n t e g a t o n fo k ^ ( ) 2 n d o de T a y l o s e e s f o 0 a ) 2 n d o de f o n J b ) N u m e c a l n t e g a t o n f o " ( ) Fxed X Cv) N u m e c a l m e t h o d s f o a n d numecal ntegaton f o t t " Note : (a) I n (), w e a c t u a l l y mnmsed t h e KL n f o m a t o n q u a n t t y f o t h e d f f e e n c e b e t w e e n t h e b e t f a c t o n s a n d t h e g h t h a n d s d e of (7.7). T h e p o g a m f o e a c h a c e s t o p s f t h a t q u a n t t y s ^ 5e-5. F o t h e d e f n t o n of t h s q u a n t t y, s e e n e x t s e c t o n. (b) I n (v), w e a g a n mnmsed t h e KL n f o m a t o n q u a n t t y f o t h e d f f e e n c e b e t w e e n t h e b e t f a c t o n s and t h e pobabltes poduced b y Gauss-Hemtan ntegatons. The pogam t e a t e s untl t h a t 59

172 q u a n t t y Is 5e-6. (Wth convegence c t e a = 5e-5 nol: mch d f f e e n c e s obseved.) Cox's t e s t s a e appled t o t e s t w h e t h e t h e most a c c u a t e method (among t h o s e n T a b l e 7.4) f o p e d c t n g p o b a b l t e s u n d e t h e Heney model s b e t t e t h a n t h e othes. The esults a e summazed n t h e f o l l o w n g t a b l e. Notce t h a t Imples t h e m o s t a c c u a t e m e t h o d (.e. c a s e (v) n T a b l e 7.4). T a b l e 7. 5 C o m p a sons a m o n g d f f e e n t m e t h o d s fo t h e H e n e y mode] b a s e d on e x a c t a b e t n M e a d o w l a n d s (50 a c e s ) Mod e l ( ) s t a) s t (H ode Taylo seesf o 32 o d e f o tt b ) 2 n d o d e f o t IJ , c) Numecal ntegaton fo. 8 3 ( ) 2 n d ode Taylo sees fo 0 a 2 n d ode f o b ) Numecal ntegaton fo n ( ) Fxed 入 F o m T a b l e 7.5, m o s t e s u l t s s u p p o t t h a t t h e m o s t a c c u a t e method 60

173 f o estmatng pobabltes s s g n f c a n t l y bette t h a n the othes a t S7* l e v e l (except () a)), even though n some cases : () b) and (), t h e absolute d f f e e n c e s o f the l o g lkelhoods a e small. The d f f e e n c e s a e e l a t v e l y moe s g n f c a n t n ⑴ a) 8 c b). The above e s u l t s depend on t h e outcomes o f hose aces but n the next: secton, we compae the d f f e e n t methods usng a cteon ndependent o f the outcomes. 7.7 Compason of pobablty estmatons usng a closeness measue Pevously, we have compaed the accuaces (as measued b y log lkelhoods) o f pobablty estmatons poduced b y t h e f x e d (A»t) model the H e n e y model a n d t h e H a v l l e m o d e l. I n t h s s e c t o n, w e a m a t c o m p a n g t h e closeness o f p o b a b l t y e s t m a t o n s p o d u c e d b y d f f e e n t models by assumng t h e Heney model Is c o e c t. T h a t s, w e w a n t t o e m p c a l l y f n d o u t w h c h model s c l o s e s t t o t h e H e n e y model. We a p p l y t h e f o l l o w n g c l o s e n e s s m e a s u e f o o u c o m p a s o n p u p o s e. a I ( 会 / ) = EE 比 ( ^ ) 一 whee tt 一 s the exacta p o b a b l t y poduced b y t h e Heney model 6

174 (usng almost exact computatos) a n d s t h e assocated p o b a b l t y p o d u c e d b y o t h e model. T h s I s c a l l e d t h e K u l l b a c k - L e b e q u a n t H y of n f o m a t o n ( h e e a f t e c a l l e d t h e KL n f o m a t o n q u a n t t y ). I t h a s t h e f o l l o w n g popetes : () K k tt ) ^ 0, () K tt ) = 0 f f 一 tt j = t j (,j = (Fo t h e poof (986).) We a d o p t t h s KL n f o m a t o n q u a n t t y a s t h e c t e o n f o t h e c l o s e n e s s of t w o d s t b u t o n s. Namely, t h e s m a l l e t h e v a l u e of I(全 tt*), t h e close w e consde the model f o t o the Heney model. The e s u l t o f compasons f o exacta p o b a b l t y n Meadowlands (50 aces) s shown n t h e f o l l o w n g table. 62

175 T a b l e 7.6 C o m p a s o n s u s n g K L nfomat o n quantty Model aveage K L s d. o f KL ( s t o d e T a y l o s e e s fo 0 ( H e n e y ) a) s t odef o " b) 2nd odef o c ) N u m e c a l n t e g a t o n f o t t " () 2nd o d e T a y l o s e e s f o 0 a) 2nd o d e (Heney) , Fxed A (v) H a v l e f o tt j b) N u m e c a l n t e g a t o n f o () I n t h e above t a b l e, ncluson of t h e H a v l l e m o d e l s t o show t h e elatve l a g e d f f e e n c e between t h e Havlle and t h e Heney models. E a c h e s u l t s a compason b e t w e e n t h e s t a t e d model w t h t h e e x a c t H e n e y model. T h e e x a c t H e n e y model s b a s e d o n a n u m e c a l m e t h o d f o co mp u tn g 0 a n d n u m e c a l n t e g a t o n f o tt (same a s j c a s e (v) n T a b l e 7.4). We c a n s e e t h a t t h e f x e d X m o d e l s c l o s e s t t o t h e H e n e y model. Besdes, () a) Is q u t e good b u t t h a t s t l l nvolves a l o t of computaton t m e w h e n c o m p a e d t o t h e f x e d A model- Hence, t h e f x e d A model s v e y close t o t h e e x a c t H e n e y model a n d v e y convenent t o u s e n p a c t c e. 63

176 7.8 C o n c l u s o n I n t h s Chapte we e c o m m e n d t o u s e a s m p l e m o d e l w h c h c a n a p p o x m a t e t h e H e n e y a n d d s c o u n t m o d e l s. F o m e m p c a l e s u l t s of c o m p a s o n s a m o n g log lkelhoods u n d e d f f e e n t m o d e l s, w e o b s e v e t h a t t h e e c o m m e n d e d s m p l e model s n o t w o s e t h a n t h e o t h e t w o whch e q u e a lot m o e computatons. I n addton, w e obseve t h a t t h e H e n e y e s t m a t e s a e n o t close t o t h e e x a c t a b e t f a c t o n s w h c h a e expected t o b e t h e best estmates of e x a c t a pobabltes. 64

177 CHAPTER EIGHT THE STERN MODEL IN JAPAN 8. I n t o d u c t o n In ths chapte, we epot the f a l u e o f the Heney model n Japan. Ths motvates us to consde anothe model - the Sten model (Sten(990)) whch assumes that the unnng tmes f o l l o w the Gamma dstbuton w t h a f x e d shape paamete,. Maxmum lkelhood estmaton of n Japan w l l also be epoted. By usng a lkelhood-based agument, we w l l show that Sten,s Gamma model w t h maxmum lkelhood estmate of s bette than both the Havlle (=l) and Heney (=oo) models. Snce pedctng complcated pobabltes unde the Sten model s a d f f c u l t job, we w l l popose a smple appoxmaton and gve numecal evdence. A theoetcal esult w l l be Included to show the exstence of a systematc bas f o pedctng complcated pobabltes f we use the Havlle model but the tue undelyng model s Sten's Gamma model- 8.2 F a l u e o f the H e n e y model n Japan In t h s secton, we t y the same condtonal analyss epoted n Chapte 5 n Japan. Recall the models ae : 65

178 logt TT^, = logt T k j = logt jk = l 0 g t TC m _ ca l o g t P C logt P =? l 0 g t P k j l l j k m ljkl (8.) w h e e a l l P ' s o n t h e g h t h a n d s d e of (5.2) a e t h e c o n d t o n a l p o b a b l t e s e s t m a t e d b y t h e H a v l l e model a n d a l l t h e l o g t s a e m u l t v a a t e. We c a n also e p l a c e a l l t h e s e P ' s b y t h e condtonal p o b a b l t e s e s t m a t e d b y t h e H e n e y model. We a l s o n e e d t h e a p p o x m a t o n f o m u l a (5.3) f o t h e H e n e y model. T h e e s u l t s a e s h o w n n T a b l e s 8. a n d 8.2. I n t h s t a b l e, w e p c k u p t h o s e a c e s wth 2 h o s e s o n l y (22 a c e s ). T h e a c e s z e s f x e d f o compason pupose because w e w a n t t o make t h e p a a m e t e estmates m o e meanngful especally f o lage q and hoses n t h e data set. 66 m o e a c e s h a v e 2

179 Table 8. Condtonal logt analyss unde the H a v l l e model paamete estmates () I(paa, est.) p a a m e t e e s t mates () I( p a a est.) Table 8.2 Condtonal logt analyss unde the Heney model p a a m e t e.246 estmates (l) I( p a a est.) 67

180 Table 8.2 (contnue) o 加 _ s e ^ o y e p g t a h I 咖 l s e l c H e 3 o e ( l o t s s c a s n t t y n t h u t n t. v t n J e h n e J l n f = t e ^ t e I. h a d 2 l, ) f s. e e 5 2 瞻 ( y 5 J h s s e t t t h 流 e e d M e n e t n t d g e a c t l e d l T p a d z n a e t a t e e t e a t - s e l. ^ u O L l C 叩 l 价 e l o H l f t a t h h a h t o a pm t g o^ w l dj b b u c n e t \ e ㈧ 3 a 吆 s t s o e e d. c t t h L e 0 l t o v o t o on d m h. s l p s g o n I no p c ^ g f l c py D s ^ g t u w c t c e a J a l g m a li e k d e l k e l l l e 咖 t e s n p a n f t V e c s a s l e I g f 8, 瓜 l e s t b d d e l. l e y f o e n t f t e e a. l k l s l e e ^ h e 办 t d m H v t t e t m u a X a b ^ o ^ ^ s. m b a ^ e q ^ H h e c e t n a s 珊 ^ - e m 2 a p 3 p e 虹 v 0 d m nt ^ V v 0 o e v t f e ^ ^ lhthefe_viheheutpome e t a 8 t l t 3 l I t y s c a t B ^ u a H E ) k n s tsat wh en nt p qv le tn K G lq.k,sts 3 ^ o ^ e %於 Jb.^otecag s n a c : ) a 让 f thkgn3c,t %^nlhgn s o ) o., 32 ) k l e o e I e/n l n ^ ( N. q = - l o n f n 对 h a g f t h B B f n s h n g postons S est.) L p a a m e t e e s t mates () q

181 T h e e s u l t Is shown n Table 8.3. T s b l S «3 Logstc analyss f o t h e fst: 3 f n s h n g postons 2 ^ A fl A 0) (,,) We h a v e n u m e c a l l y s h o w n t h a t ( s e e C h a p t e 7 t h e (X,t) m o d e l s a good a p p o x m a t o n of t h e H e n e y m o d e l f o e s t m a t n g n whee, jk 入=0.76 and t=0,62. We can see t h a t n Japan, close to the values o f one. T h e e f oe, t s not s u p s n g t h a t the H a v l l e model w o k s b e t t e t h a n t h e Heney model f o p e d c t n g the p o b a b l t e s f o the second and t h d f n s h n g odes. 8.3 F t t n g t h e S t e n m o d e l Sten's Gamma model (Sten (990)) s motvated b y consdeng a competton n whch n playes, scong ponts accodng t o ndependent Posson pocesses, a e a n k e d accodng t o t h e t m e u n t l ponts a e scoed. Thus should be a n ntege unde t h s assumpton. Whethe t h s assumpton s easonable o not w h e n appled t o h o s e - a c n g poblem s a n open queston- But w e can consde It as a n a l t e n a t v e model t o the H a v l l e and Heney models. L e t the 69

182 u n n n g tme o f hose G a m m a C, ^ n d e p e n d e n t l y o = W ) f(f) > w h e e s pedetemned and c a n b e e s t m a t e d f o m tt. (o t h e b e t f a c t o n, P^), We m a y t y t o e s t m a t e t h s b y c o m p a n g t h e log lkelhood : y I n, w h e e [23] d e n o t e s t h e 3 t o p h o s e s n a c e, L [23 w t h d f f e e n t v a l u e s of. T h e e s u l t f o J a p a n e s e d a t a s s h o w n n T a b l e 8.4. T h e c o m p u t a t o n s a e d o n e b y u s n g G a u s s - L a g u e e n t e g a t o n f o t h e S t e n model a n d G a u s s - H e m t a n n t e g a t o n f o t h e H e n e y m o d e l (.e. =oo). F o t h e S t e n model, w e n e e d t o f n d 0 f s t b y solvng t h e f o l l o w n g e q u a t o n : ^ : 广J As usual, e )] H s 0 G (t 0 ) s s s the ^ cumulatve dstbuton functon a s s o c a t e d w t h g (X 0 ) I n f a c t, f o n t e g e, t h e above P h a s s s I a c l o s e d f o m n t e m s of 0- H o w e v e, t m a y n o t b e p a c t c a l u n l e s s n and a e small. 70

183 Table 8.4 Log lkelhood values unde the Sten model f o Japanese data log l k e l h o o d (.e. H a v l e ) o (.e. Heney) Fom the above table, the log lkelhood s maxmzed when =4 Thus the Gamma dstbuton w t h =4 s a bette dstbutonal assumpton of unnng tme n Japan. We may also f t the Sten model n Hong Kong and Meadowlands and the esults ae shown n Table 8.5 and 8.6 espectvely. Table 8.5 L o g l k e l h o o d v a l u e s u n d e t h e S t e n model f o H o n g K o n g (89) d a t a log l k e l h o o d o (Heney)

184 Tab e 8. 6 Log l k e l h o o d v a l u e s u n d e t h e S t e n model f o Meadowlands data log lkelhood S (Heney) We can see that =oo (.e. the Heney model) appeas to be the best n both data sets. We may wonde why the dstbuton of unnng tme does not unvesally hold. We cannot f n d any s u f f c e n t eason f o t h s. One d e a Is that the allocaton of the puse to the top 5 hoses s dffeent n d f f e e n t acetacks. We do not have t h e puse n f omaton f o Meadowlands data but we have collected t h s n f omaton f o Hong Kong and Japanese data n Table

185 Table 8.7 Allocaton o f puse n pecentages F n s h n g hose Hong K o n g Japan st nd d th th F o m the above table, the pecentages o f t o t a l puse a e hghe f o 3d, 4 t h & 5th hoses n Japan than n Hong Kong. We suspect t h a t the w a y o f t a n n g o f hoses and jockeys n Japan s not the same as n Hong Kong. Moe clealy, n Japan, the f a v o u t e hoses may t y h a d t o get n the top 5 postons even f they ae f a away f o m the leadng hoses. Ths w l l ncease the condtonal pobabltes o f f n s h n g 3 d f o favoute hoses. Ths dea s smply a suspcon only and, o f couse, thee may be othe moe easonable f a c t o s w h c h w l l a f f e c t the unnng tme dstbutons o f hoses. 8.4 A s m p l e a p p o x m a t o n o f t h e S t e n m o d e l The (A,t) method poposed n Chapte 7 s smple enough t o apply t h e Heney model n pactce. In t h s secton, we w l l see t h a t the same dea can be appled f o the S t e n model. We defne : 73

186 , G() - ( InlTt, / j tt ). ) ln(7^/7 ^) (8.3), G() = Jk (7 () Ijk /, () 7. J IJI ln(7 / t t } () w h e e tt^ s P(hose w n s a n d h o s e j f n s h e s 2nd) u n d e t h e S t e n model w t h s h a p e p a a m e t e. S m l a l y f o tt (). If w e c a n jk assume t h a t t h e above t w o values a e close t o t w o constants, denoted b y X G ( ) a n d T G ( ) espectvely, t h e model f o appoxmaton of t h e S t e n model s : k t G() G() () jk.g() s JU (8.4) G() w h e e tt ' s a e e s t m a t e d b y t h e w n b e t f a c t o n P *5 入G() and TG() can be estmated by the mean of 入G() and TGC based on a j jkl lage numbe o f aces o by the a t o of maxmum lkelhood estmatos (see Chapte 7 f o detals). Hee, we choose the f s t method because the () second one eques to compute tt f o a l l combnatons and f o each ace. We have lage numbe of aces n Japan and s vayng and thus, the second method w l l be too tedous. 74

187 The summay values o f A G() and T G() based on 583 aces In Japan a e shown n Table 8.8. Table 8.8 Summay values o f X G ( ) and 丁G() xg() T s.d. mean mean G() s. d ( N o t e : I n t h s table, w e have set,j k e q u a l t o t h e h o s e s f n s h n g n t h e t o p t h e e postons n each a c e a n d s vayng f o X G < ) a n d t G ( ) n (8.3).) Jl ljkl F o m t h e ab o v e t a b l e, t h e s t a n d a d d e v a t o n s a e q u t e s m a l l, n g e n e a l. H e n c e, w e e x p e c t t h e m e a n v a l u e s a e good a p p o x m a t o n s t o t h e 入 G ( ) and T G ( ) f o d f f e e n t combnatons o f hoses n j ljkl d f f e e n t aces. We a l s o compute the summay s tat s t c s o f K L n f o m a t o n quanttes f o d f f e e n t to compae the t u e (obtaned b y n u m e c a l ntegaton) w t h the above f x e d 入G() method. Also, the K L n f o m a t o n quanttes f o compang t h e t u e tt^ w t h those 75

188 pedcted by the Havlle model s teated as a contol f o compason. Table 8.9 Compason between f x e d 入G() model and the Havlle model usng K L n f omaton quantty (500 aces) Ave. K L A G() Ave. K L sd. K L Havlle sd. K L Clealy, b y compang t h e KL nf omaton q u a n t t e s above, o u 入G() method s much accuate than the nave Havlle model f o pedctng the complcated pobabltes based on the Sten model. Moeove, we compae the log lkelhood values of Sten models usng numecal ntegatons (.e. f o m Table 8-4) w t h ou f x e d (X G(), TG()) model f o pedctng IJK n the followng table. obseve that the log lkelhood values based on two methods do not have bg dffeences. 76

189 Table 8.0 Compasons o f l o g l k e l h o o d v a l u e s f o Japanese data numecal ntegatons (入G(), t g < ) ) ( H a v l l e ) -8977, o (Heney) 8.5 A t h e o e t c a l e s u l t f o t h e Havlle f x e d and compason between t h e S t e n models I n t h s secton, unde the assumpton o f the Sten model w t h paamete, the d f f e e n c e n estmatng the condtonal pobablty o f hose j f n s h n g second gven that hose fnshes f s t by the H a v l l e and Sten models w l l be Investgated. S m l a to Chapte 5, we w l l study the f o l l o w n g d f f e e n c e : = 小 tt. 小 - t (8.5) f o j = a and b, whee a and b ae hoses chosen such that : 77

190 and s based on the Sten model w t h shape paamete. Note t h a t ECT^) = /0^ f o = l, 2, " n a n d t h u s, E(T ) = m n E(T ) a n d a s ^ s E(T ) : m a x E(T }, b s ^ s so t h a t a s t h e stongest hose and b s t h e weakest hose as m e a s u e d b y t h e m e a n u n n n g t m e s. I n t h e f o l l o w n g, w e u s e g (v) t o d e n o t e t h e p. d. f of V w h c h s a gamma a n d o m v a a b l e w t h s h a p e paamete and scale paamete L Its assocated c u m u l a t v e d s t b u t o n s G (v). To s t u d y (8.5), t h e f o l l o w n g lemmas a e e q u e d. L e m m a 8. Let u,v a n d w b e non-negatve functons. I f, n addton, both u and v / w a e non-nceasng, w e have : Suv S v Smlaly, f u s non-nceasng and v / w s non-deceasng, t h e n w e have

191 Poof Ths esult S uv J 丄u v by lemma 5. ( s n c e l / u s non-deceasng) S m l a l y f o t h e second p a t. L e m m a 8.2 v g (v) e (v) ^ whee e{v) = f o v > 0» = l, 2, 3 - G (v) Poof ehv) _ [ - G ( v ) ] g ( v ) ( - v ) + v g ( v ) 2 [ - G ( v ) ] 2 N o t e t h a t g^tv) = (二I!,l e a n d 卜 ⑷ = e [ q * T o s h o w t h a t e ' t v ) 彡, t s equvalent to show the f o l l o w n g nequalty : 79

192 (-)! (-v) (-l)! (8.6) To pove (8.6), we use mathematcal nducton on. It s obvous that when =l, the above w l l become an equalty. Assume (8.6) holds f o a patcula, we have to show that f o +, (8.6) s t l l holds, that s, we have to pove the followng nequalty (+l-v) - q RHS = I I ^ I 2 [ [- 厂 一 上 + 2 [ J] (-v) + (-D! [C-)! the nducton assumpton. I LHS = [ f ](+l-v) (-v) E Theefoe, 80

193 RHS - L H S <5=0 [ ~ ] 2 ( v - ) + v [ ] 2 [ - ( ¾ 2 ]! (-)! = [ - E,0 ^ q! q ( 卜 - )!, 2 [llxlli] - + v J[ L _ f (-D! (. I ) I t s c l e a t h a t t h e l a s t expesson ^ 0 f v ^. T o show t h a t t h e l a s t expesson 2 0 f o v >, t s e q u v a l e n t t o s h o w t h a t t h e followng nequalty holds f o v > : ( E q=o $ ) [ ( v - ) 2 + v] ^ ^ v(v-) (-)! (3.7) T h o u g h (8.7) a p p e a s t o b e s m p l e t h a n (8.6), w e s t l l c a n n o t e a s l y s e e t h a t (8.7) h o l d s. We n o w u s e n d u c t o n o n a g a n f o (8.7). When = l, LHS o f (8.7) = v 2 - v + a n d RHS = v 2 - v a n d t h u s t h e a b o v e n e q u a l t y h o l d s. Assume (8.7) h o l d s f o a p a t c u l a v a l u e o f, w e h a v e t o s h o w t h a t f o +, t h e f o l l o w n g n e q u a l t y h o l d s : ( [ Z - ) [(v--l) 2 + v] ^ 8 vcv--l)

194 L H S o f t h e above n e q u a l t y = ( [ ~ Y q q=o - ) [ ( v - ) + v - 2 ( v - ) + +! ~ ^ - l q [ ( v - - l ) 4- v ] ) [ l - 2 ( v - ) + t(v--l) 2 4- v! v(v-) + ( E (-)! q=o b y t h e Inducton assumpton. Thus, LHS 一 - - l q v(v-) - ( [ ) [2(v-)-] + [(+ 2- v ] (-) = ( 2 + l ) - (J ^! q=o ) [2(v-)-l] T h a t s, w e have t o show t h a t : (2+l)! 2 l q=o t ^ ) [2(v~)-l] (8,8) Agan, n d u c t o n s a p p l e d t o (8.8) o n. When =l LHS = 3v > RHS = 2 v - 3. Assume (8.8) s t u e f o a p a t c u l a, w e h a v e t o s h o w t h a t t h e n e q u a l t y a l s o h o l d s f o +,. e. [2(+l +l 之 E ) [ 2 ( v - - l) - l] (+)! q=o + 82

195 R H S o f the above nequalty )[2(v-)-l-2] [2(v-)-l -2 ( E ^ - ) [2(v-)-l]- [2(v-)-l] (2+l) - 2 by t h e nducton assumpton. A ly ( V 2( Now, t s easy t o see t h a t LHS - RHS ^ +2 ^ (+)! q=o Hence, b y nducton, (8.8) h o l d s f o a l l =l 2. so do (8.7) and (8.6) a n d t h u s, e (v) ^. L e m m a 8.3 : Defne t h e followng functon : J K(v 0 ) = g (v0 / 0 ) 0 / 0 J s ( 州 / s关j j 0 s / 9 E t tg (v0 / 0 W 二 ve ) JlsV /V (8.9) w h e e G (v) = - G (v). Then we have the f o l l o w n g e s u l t s 83

196 K(v 0 ) Is non"-nceasng n v and a K(v; 0 ) s n o n - d e c e a s n g n v? b whee 0 = Max 0 a s 0 b and = Mn 0 ^ s Poof : C o n s d e t h e d e v a t v e of K(v; 0 )wth e s p e c t t o v : d K :V; d v W [ E v s^j 0 3 V s^ j e s / e S ^ / 9 s共 J 0 /0 n ve / 0 w式 2 w, -0 /0 w^ij w G (ve / 0 ) w 丨 g (V0 / 0 w G (V0 / 0, )! M v 0 /? G (v0 / 0 ) h^te" j t S, s t -0 / 0 - U, t G ( y e / 0 ) I w 9 V } U (ve / e G (V0 / 0 ) / 0 g (v0 / 0 ) / V e / 0 g (ve / 0 ) J J + g (V0 / 0 ) ( v d (ve / 0 ) 3 G (ve / 0 ) s s t t^ts 84 H (V0 / 0 } G (v 0 / 0

197 0 /0 j j w* j w s _ 0 / 0 V0 / 0 卜 e/ 9 s^ij j w ^ l j J G (ve / 0 n G (ve / 9 ) g (ve / 9 j w / e g ( v e / 9 ) n G (vg / 0 ) s a s s t ve/ 9 g ( v e/ 6 j E { e/ e 竺 I s s萁 j e /9 &g (ve /g ) t t ^ 0 /0 e/ 0 ) I g (ve / 0 ) n g (ve / e j s 0 /0 g (ve/ 0 )?= s t ^ + 0 /0 ve/ 9 0 / 0 G (v0 / 0 ) w g (ve / e ) n s g (V0 / 0 ) 0/ 0 v e/ 9 g ( v 0/ 0 G (v0 / 0 ) G (v0 / 0 ) /0 g (ve /0 ) n G (y0 / 0 & w* \ /0 g (Ve s^ /Vt5lsVVVe (v0/ 0 g (v) whee e(v) - G (v) 85 ) - e ( v 0/ 0 ) s I t^is t

198 When j=a, d 0 & ) d 0/0 a W{ 2 W s^ia V 0 s / 0 W g (ve / 0 ) T I G (ve / 0 ) a I w其 l a w {es I s笨 a ^ 0 0 / ( - V 0-8 ("V by the mean value theoem whee v 0 (v0 / 0, v0 / 0 ) s a ^ 0 by lemma 8.2 On t h e o t h e h a n d, When j = b, d K(v; a ) f ^ 二 S^b 0 / 0V W J (ve / 9 ) e / e s b s W n g t 六 s (ve / e V s^b ' e'cvj by the mean value theoem whee v (v0 / 0, v0 / 0 ) b y lemma t I

199 We have shown t h a t d K(v; 0 ) a d v : d K(v; 0 ) 彡 0 and b ^ d v hence t h e e s u l t f o l l o w s. T h e o e m 8. TC n丨 a, I _ and 彡0 T w h e e a and b a e c h o s e n s u c h t h a t 0 a = Max 0 and s 0 b =M n 0 s Poof Consde t h e d f f e e n c e (8.5), tt n 丨-- -~ = ji 一 % [ tt n 一 Ij 7 ^ lt ] l G(u0/0^) du 0 oq oo g (u)仏 [ 一 ( /0 ) du - J n Jd) - tt J w h e e tt = P(T < M n {Ts } j⑴ J 87 g j u ) 爲 U-GJuQ^/Qp d u

200 .e. the pobablty of hose j wns f hose I s emoved f o m the ace. Theefoe, JCl) Defne g, 6 j = tt - JCl) - 7 T h u s, t s u f f c e s t o show t h a t g y ] s 0 a n d j(n Consde : ) 88

201 J o T I w垆 j G ( u 0 / 0 ) g (u) d u oo E +n t ^ s^j ^ 0 w j 一 G s (U0 /6 ) g. (u) du t s 03 G (u0 / 8 ) J T I j T I w9= j G ( u 0 / 0 ) g (u) d u G (u0 / 0 ) T I s关j J 0 s (v0 / 0 ) 0 / 0 j j, s垆 j s n t^s t 丨 (v0 / 0 ) 0 / 0 s s n 万 Cv0 /0 ) dv w^j (V0 / 0 ) 0 / 0 j G Cue / 0 ) g ( u ) d u 9= s (v0 / 0 ) 0 / 0 s s萁f w s w G (v0 / 0 ) dv t G (v) 3 G (ve / 0 d v w 4 (v) T I s w G (v0 / 0 ) d v t b y change o f vaables usng v = u0 / 0 n the n u m e a t o, and v = ue / 0 n the d e n o m n a t o. ^ 0 when j=a b y usng Lemma 8.3 togethe w t h Lemma 8. 之 0 when j=b Hence, g a 彡 0 and g. 之 0 o and t h e theoem s poved. 89

202 Note t h a t w h e n =l the S t e n m o d e l e d u c e s to t h e Havlle model and t h u s n t h e a b o v e t h e o e m, t h e t w o nequaltes wll b e c o m e t w o equaltes. Smla t o t h e t h e o e m s t a t e d n Chapte 5, t h s t h e o e m means t h a t f t h e u n n n g t m e s s a t s f y t h e a s s u m p t o n of t h e S t e n model ( w t h p a a m e t e =2,3,...), t h e H a v l l e m o d e l w l l o v e e s t m a t e t h e c o n d t o n a l p o b a b l t y of t h e m o s t f a v o u t e h o s e f n s h n g second a n d u n d e e s t m a t e t h e condtonal pobablty of the longshot f n s h n g second. We h a v e s h o w n t h a t a s y s t e m a t c d f f e e n c e a p p e a s ( e t h e oveestmate o undeestmate t h e pobablty) w h e n w e use t h e Havlle model t o compute t h e odeng pobablty w h e n t h e t u e u n n n g t m e d s t b u t o n s N o m a l o Gamma. T h e s e t w o t h e o e m s ( t h e o e m 8. a n d t h e o e m 5.) a e b o t h p a c t c a l l y a n d t h e o e t c a l l y n t e e s t n g s n c e t h e H a v l l e m o d e l s m o e commonly u s e d n p a c t c e. We m a y c o n s d e N o m a l a s a p a t c u l a c a s e of Gamma a s w h e n > oo. Gamma > Nomal. I t may also b e nteestng t o compae Sten() w t h Sten(-l),.e. w e thnk t h a t t h e followng conjectue should hold : 90

203 Conjectue () _ la (-) () and & 2 n b (-l) b f o a n y a n d = 3,4,... whee (t) 0 a = Max 0 ^ s s^ 0 = Mn 0 b s s^i» P ( h o s e w n s and j f n s h e s second when S t e n ( t ) s assumed) G ( u 0 ( t ) / 0 ( t ) ) g (u) t j fo t =, -l T s ^ j [-G t (u0s (t) /e (t) )] d u J and 0⑴ s a e obtaned by solvng the f o l l o w n g system o f nonlnea k equatons : I ⑷ J k ^ ( u e ^ / e ^ ) ] du f o t =, - l a n d k = l,2 n. T h e a u t h o h a s n o t b e e n a b l e t o p o v e t h s c o n j e c t u e. But e x t e n s v e n u m e c a l e s u l t s ( f o =3»... > 8 I n J a p a n e s e d a t a ) v e f y that ths conjectue holds. Although ths conjectue Is of theoetcal nteest, t s not vey pactcally mpotant snce people wll n o t appoxmate Sten() w l l n g t o u s e t h e S t e n model. 9 b y S t e n ( - l ) once t h e y a e

204 8.6 C o n c l u s o n Ths chapte demonstates the applcaton of the Sten model n Japan. Thus the Sten model seves as an altenatve model f o p e d c t n g complcated odeng pobabltes to the Heney model. As no one model s consstently bette than the othe n d f f e e n t acetacks, w e should nvestgate t h s ssue f u t h e, f possblethee should be some eason f o t h s nconsstency. But we can s t l l apply dffeent models n dffeent acetacks. A. smple appoxmaton of the Sten model n computng the complcated o d e n g pobabltes s poposed. We have also shown that the H a v l l e model w l l have a systematc bas f the unnng tme d s t b u t o n s gamma athe than exponental. 92

205 CHAPTER NINE STUDY OF BETTING STRATEGY 9. I n t o d u c t o n In t h s chapte, we w l l apply a bettng system 一 D. Z ' s system on ou avalable data sets. F a l u e to ean p o f t n Hong Kong motvates us to thnk of mpovements. By usng the f x e d (A,t) model poposed n the pevous chapte, the obseved p o f t s ae geneally nceased. Bettng stategy s thus an mpotant applcaton o f the f x e d (X,T) model. 9.2 D e s c p t o n o f D. Z ' s system Hausch, Zemba and Rubnsten (98) (HZR) develop a system w h c h s capable o f explotng dffeences n effcences Impled by d f f e e n t bettng pools. It s called D.Z's system. Assumng the w n pool poduces accuate estmates, the system seaches f o undebet hoses n place and show pools. Testng the system at Santa Anta (Los Angeles) and Exhbton Pak (Vancouve), pue p o f t was epoted. Moe empcal evdence o f makng p o f t s can be found n Zemba & Hausch (987). The detals o f D. Z ' s system ae explaned below. 93

206 9.2, Selecton of hoses Usng the smple Havlle model, D.Z's system eplaces the wnnng pobabltes by the bet factons and estmates moe complcated pobabltes such as t f ecta pobabltes 芤 k. We f o l l o w the temnology n U.S. bet types that place bet pays o f f f the hose fnshes n the top two and show bet pays o f f f the hose fnshes n the top thee. Followng H Z R (98), the expected e t u n f o m a $ bet to place on hose s gven by E (Xp) = Z j^ (TT U + TI ) Ret j l U (9.) w h e e 7^= P (hose f n s h e s st a n d h o s e s j f n s h e s 2nd) u n d e t h e H a v l l e model, Ret U = P l a c e e t u n on h o s e p e d o l l a f h o s e s a n d j f n s h f s t a n d second, e g a d l e s s of t h e o d e. D. Z ' s s y s t e m decdes t o h a v e a p l a c e b e t on h o s e If E(X^) I s a t l e a s t a c e t a n value, a (>). Smlaly, w e c a n e s t m a t e t h e e x p e c t e d e t u n fom, a $ b e t t o show on h o s e a n d p u t a b e t f E(Xp ^ a (>). Assumng t h a t t h e e s t m a t e d e x p e c t e d e t u n s a e a c c u a t e a n d f t h e w n b e t s n e a l y e f f c e n t, t h s selecton p o c e d u e m e a n s t h a t w e should look a t t h e m a k e t n e f f c e n c e s of p l a c e a n d show b e t s. Note t h a t n some a c e t a c k s, e t h e p l a c e o 94

207 s h o w bet s avalable but not both n a ace. It s qute clea that t w o poblems m a y exst : () T h e f a v o u t e - l o n g s h o t bas m a y affect the accuacy of the bet factons, () H a v l l e ' s f o m u l a s not accuate enough Optmzaton o f bet amounts: A f t e selectng the hoses w h c h ensue a mnmum expected e t u n, D. Z ' s system maxmzes the K e l l y c t e o n w t h espect to the bet amounts put on the selected hoses. The objectve s t o maxmze the a t e o f gowth of e t u n by maxmzaton o f expected log u t l t y o f wealth. A lot of hghly theoetcal studes and some e m p c a l studes (especally n stock makets) fo the K e l l y c t e o n and t s compason w t h Makowtz's mean-vaance c t e o n have been done (See K e l l y (956); Beman (960); Thop (969,97); R o l l (973); Mae, Peteson & Wede (977) and Gaue (98), etc.). To maxmze the K e l l y cteon, D.Z's complcated nonlnea pogammng as fo l l o w s. 95 system nvolves

208 Defne 厶 = s e t o f selected hoses, P I 广 O g n a l p l a c e bet made by t h e populaton on hose, Sh! = O g n a l show bet made by the populaton on hose, PI = Z PI, S h = E Sh, o p l = o p t m a l b e t amount t o p l a c e on h o s e ( t o b e d e t e m n e d ), osh^ = o p t m a l b e t a m o u n t t o s h o w on h o s e (to b e d e t e m n e d ), t = tack take, T^k= e s t m a t e d t f e c t a p o b a b l t y b a s e d on H a v l l e m o d e l, W = n t a l w e a l t h b e fo e t h e a c e. o The nonlnea pogammng poblem s Maxmze E E S ^j^k n In U jk (9.2) jk whee U ( l - t ) (PI 么oplj - P I, PI *4 o p l ^ o p y = k 2 opl opl o p l + PI o p l + PI (-t) ( S h + E o s h ) - (Sh + S h + S h + o s h + o s h 厶 I osh osh osh + S h osh + S h 0 - E ^ jk e d osh opl ^ j T ^ 厶 96 k osh osh + Sh I + osh )

209 subject t o t h e f o l l o w ng constants Y ( o p l, osh!) s W opl opl, osh In wods, U Uk 之 0 f 厶 e p e s e n t s t h e w e a l t h a f t e b e t t n g on t h e c u e n t a c e w h e n h o s e w n s, h o s e j f n s h e s second a n d h o s e k f n s h e s thd. T h s p o b l e m usually t a k e s much t m e t o solve f o each a c e n p a c t c e. HZR suggest a n o t h e s m p l e p o c e d u e u s n g a s e t of e g e s s o n e q u a t o n s (See HZR (98) f o d e t a l s ). If t h e p o b a b l t y e s t m a t o n s e x a c t (whch s n o t e a l l y t u e ), t m a y b e of I n t e e s t t o n o t e t h a t t h e w e a l t h h s t o y u n d e D.Z's system f o m s a sub-matngale stochastc pocess. T h e o e m 9. Unde D.Z's system, t h e w e a l t h hstoy s a sub-matngale stochastc pocess. Poof L e t W b e t h e w e a l t h a f t e t h e q t h a c e u n d e D. Z ' s system, q Also let R q and B ql (2:0) b e e s p e c t v e l y t h e n e t e t u n p e one u n t b e t a n d t h e optmal b e t a m o u n t d e t e m n e d b y t h e s y s t e m o n t h e 97

210 hose I n a c e q (Net e t u n hee means e t u n pe u n t bet mnus ). Note t h a t t h e numbe o f bettng oppotuntes unde the selecton pocedue o f D. Z system s less than o equal to the t o t a l numbe of hoses n t h e ace. q Then, W = Wo + ^ q = n E R B whee W s t h e n t a l w e a l t h, = f o f n t e q =,2 and n = the numbe o f hoses n ace. D e f n e ^ be a c-feld geneated by t h e w e a l t h h s t o y up t o peod q q,.e. f = c(w W ", W ) 0 q q We f s t p o v e t h e f o l l o w n g I n e q u a l t y h o l d s. () E ( W q ) < o Note t h a t R 一 f = q v q V q < o because : Mql f t h e hose f n s h e s st, 2nd o 3 d - othewse (Wthout loss o f genealty, assume t h s bet s a show bet.) w h e e M (>0) s anothe andom v a a b l e snce t h s amount depends on q w h c h othe hoses f n s h n t h e top t h e e postons. It see t h a t : (l-t)sh < q < o 3 Sh whee S h s t h e t o t a l show bet made b y t h e p o p u l a t o n o f b e t t o s on h o s e a n d S h = S Sh 98

211 and thus, R q ^M Also, note t h a t R < oo. q p J W f o p>q. q When Hq=l W = W + V R B, t h e n * o ^ u 叫W ^W + E ( I R u B} ^ W + I W E q R j 0 0 < snce B u ^ Z B s w U j J 00 Now, a s s u m e E ( W j ) < oo. Then, E( W q + 丨 w q ) ^ w q + ^ W q. snce B EE( R+uBq+up W q E ( R q + u ) ^Y B ^W q+, ^ q+.j q Theefoe, E C W q + ) =E [ E ( W q + W J $ E q W q ^ E C I W J ) E( Rq+M ) = E( Wq ) < o I + E E( Rq + j l b y the assumpton. 99 )] 0 0

212 Thus, the poof of ths pat completes by nducton. The second p a t s : ( ) E ( W 矛 = E ( W I W q cj- q ) <3-2^ W q- fo q =, 2,,.. To p o v e t h e a b o v e, n o t e t h a t : W = W q + V R B q- q q. theefoe E(W I W q ) =W q- q- ) B, q q, because E(R ) 之 0 V, q unde t h e selecton method o f D.Z s q system. H e n c e, t h e w e a l t h h s t o y {W } s a s u b - m a t n g a l e s t o c h a s t c q pocess. T h u s, W ^ E(W ) ^ ^ E(W ) I f a b e t t o uses "andom bettng" (a t e m used b y Hausch, Zemba & R u b n s t e n (98)) b a s e d on t h e s a m e b e t a m o u n t d e t e m n e d b y D.Z's system, t h a t means t h e b e t t o b e t s o n e v e y hose w t h p o p o t o n o f m o n e y d e t e m n e d b y t h e b e t f a c t o n s of g e n e a l p u b l c, o t h e b e t t o chooses a h o s e w t h p o b a b l t y e q u a l t o t h e assocated b e t f a c t o n, then : 200

213 E C W 歹 = E ( W W } q' q- q q- (-tj S h - S h - Sh - Sh 2 m 3 Sh Sh H ) E s = W m q- h [ 凡 shm - t 7 B ^ W ( a supe-matngale). q q- whee t = tack take, s u b s c p t [] d e n o t e s t h e h o s e w t h f n s h n g o d e L Thus, W 2 E(W ) > 2 ECW ) ^. T h e e fo e, t h e e x p e c t e d w e a l t h u n d e D. Z ' s s y s t e m s n c e a s n g ove t m e a n d D.Z's system s, on aveage, always b e t t e t h a n t h e andom b e t t n g povded t h e estmated expected e t u n s a e hghly a c c u a t e. T h s, of c o u s e, e q u e s good e s t m a t o n of complcated pobabltes A p p l c a t o n s of D. Z ' s s y s t e m I n t h s s e c t o n, w e t y D. Z ' s system n Meadowlands (U.S-) a n d Hong Kong e s p e c t v e l y. The f s t s t e p s t o s e l e c t t h e h o s e s t o b e b e t o n. R e c a l l t h a t t h e e x p e c t e d e t u n f o m $ b e t t o p l a c e on h o s e 20

214 I s E (Xp) = Z Z (TI + U TI ) Ret Jl U whee R e t = P l a c e e t u n on h o s e p e d o l l a f h o s e s a n d j f n s h f s t a n d second, e g a d l e s s of t h e o d e ( d f f e e n t f o m county t o county ) (l-t)(fl+l) - (l+pl^pl^) + INT [ * 0 ] / 0 In U.S. 2 (+P ) (-t)(p+) - ( + p ^ p y = 来 00 ] / 00 + ound [ 2 (+P^ n Hong Kong. PI = t o t a l d o l l a amount bet to place on hose j, j PI = Z PI and t = t a c k take. J J The f u n c t o n INT means tuncaton and ound means oundng. The dffeences between the above etuns ae due to the d f f e e n c e n the use o f Beakage. The t a c k take Is Hong Kong and Meadowlands (U.S.) espectvely- S m l a l y, we can obtan the fomulas f o m a $ bet t o show on hose (Le. E(X^)) f o d f f e e n t countes. H Z R suggest to bet place on 202 hose I f E (X^) ^

215 s h o w on hose If E(X^ ^ a. The cutoff value c t Is chosen "subjectvely". I n H Z R (98), "Intuton suggests that Santa A n t a w t h t s l a g e bettng pools would have moe accuate estmates o f the t h a n would be obtaned a t Exhbton Pak. Hence postve p o f t s w o u l d e s u l t f o m lowe values o f oc.". Hence, they use a=.20 f o E x h b t o n P a k and a=l6 f o Santa Anta. It s known that the pool szes p e a c e n Hong Kong a e geate than those n U.S.. We should f o l l o w H Z R t o use.6 f o Hong Kong and Japan but we have a c t u a l l y t e d moe values o f a (.2,.6 &.20) In Hong Kong and Meadowlands t o see the senstvty o f the esults. Some v a a t on s o f D. Z ' s system a e : () F o the selecton u l e o f D. Z ' s system, n addton to the constant t h a t expected e t u n a a, w e m a y h a v e a n a d d t o n a l p o b a b l t y e s t c t o n : P f h a v n g a p o f t ) 之 p n o d e to sceen out longshots and thus t s s a f e. I n the f o l l o w n g analyss, we mpose one o f the two sets o f e s t c t o n s In some cases : at. P ( f n s h e s st o 2nd)之 0.0, and P ( f n s h e s st, 2nd o 3d) ^ 0.20 b. P ( f n s h e s st o 2nd)之 0.20, and P ( f n s h e s st, 2nd o 3d) These sets o f e s t c t o n s sceen out t h e longshot hoses so tjat even f the pobabltes nvolved n the longshots a e not accuate enough, they w l l not be selected f o ou optmzaton. Note t h a t b s moe e s t c t e d t h a n a. 203

216 () Fo the estmaton of tfecta pobabltes usng w n bet factons t h e H a v l l e model I s b e n g u s e d. Howeve, f o m t h e e s u l t s o b t a n e d n t h e p e v o u s c h a p t e s t h e H a v l l e model I s a l o t w o s e t h a n t h e H e n e y model. I n c h a p t e 7, w e h a v e su g g e ste d t o u s e t h e f x e d X a n d x method f o convenence a n d w e have empcally s h o w n t h a t t h s s m p l e m e t h o d s a s good a s t h e H e n e y m o d e l. Hence, f w e u s e t h e f x e d A (=0,76) a n d x (=0.62) m e t h o d f o t h e s e l e c t o n o f h o s e s a n d o p t m z a t o n of b e t amounts we e x p e c t t o h a v e m o e accuate values of optmal b e t amounts snce t h e pobablty estmaton s mpoved. () We m a y combne () a n d () above. T h a t s, p o b a b l t y e s t m a t o n s t o b e mpoved f s t a n d t h e n n t h e s t a g e of s e l e c t o n o f h o s e s, t h e mpoved p o b a b l t e s a e u s e d t o e s t m a t e t h e expected e t u n s and t h e pobabltes of havng p o f t f o d f f e e n t hoses. Fnally, t h e mpoved pobabltes c a n b e used f o t h e o p t m z a t o n pocedue^ Some e m p c a l e s u l t s f o t h e above I d e a s a e s h o w n n T a b l e 9. f o Hong Kong (85-89, 229 a c e s ). Moeove, w e s h o w t h e e s u l t s w t h o u t e n t a l z n g t h e c a p t a l e a c h y e a f o m 85 t o 89 n T a b l e A p a t f o m t h e f n a l wealth we a l s o s h o w t h e t o t a l b e t a m o u n t s a n d mnmum captal dung t h e w e a l t h h s t o y n t h e tables. I n t u t v e l y, m n m u m c a p t a l s a n n d c a t o of s k a s s o c a t e d w t h t h e bettng method. In all ou analyses, w e maxmze t h e objectve f u n c t o n usng a numecal method a t h e t h a n t h e egesson 204

217 appoxmaton poposed by Hasuch, Zemba & R u b n s t e n (98). Table 9. (a) D.Z's system and ts vaaton n Hong K o n g fo Model Yea fo odeng pob. o c =.2 o g nal $ o (30597) 262] 30 (30685) I (35995) [407] 659 (350262) [340] (543389) [965] 2450 (545468) [ (370076) [45850] 5843 ( ) [48646] (98704) [ (993427) [62469] (323674) [ (285259) [28404] 4034 (26650) [ (25800} I ( 8 6 2} [ (35478) [8780] (4476) [ (309 30) [052] {27564) [94402] (6399) [ ] (23675) [50 409] (98620) 7336] (70205) [22022] (88333) [52843] C 89963) [80892 TJ [ %J t 6 3 l f. 2 TJ l x u o [ 7 2 o t 5 c 205 / ( x, t ) u 85 f% Fxed HK $ f% Havlle w t h pob. e st c t I o n K H HK $ w t h pob. e st ct o n

218 Table 9. (b) o g nal H K $ F x e d w t h pob. estcton H K $ w t h pob. e s t cton H K $ ( [2393] 5233 (25544) [2393] (24394) [2393] ( [2285] 2739 (36464) [ ]] [ 3637 (339943) [[2260] 2260] (426364) [0084] 2555 (43429) [0503] 2555 (43429) [0503] (58 650) [47565] ( ) [50375] (620387) [ (8844) [7835] (8844) [ ] (368735) [84429] (294752) [ t ( ) (87972} [ (20439) [20384] (38965) [ (94235) [ (22807) [76947] (40682) [73523] 9047 (3638} [9046] (26028) [60892] (74624) [84366] (5637} [94627] (30762) [ (33836) [ ( x, t ) (9975) [74454]

219 Table 9. (c) w t h pob. e s t ct: o n $ K H HK $ 6 8 (203573) [2478] 6 8 (203573) [2478] 4562 (96730) [2478] (253030) [0078] 8869 (236765) [8869] 233 (257733) [233] (35665) [49649] (360738) [ ] (360738) [5692] (428535) [7464] (449) [77603] (44408} [77 603] (404802) [60997] \J u 2262 (404802) [60997] (434480) [68093] (37 624) [ (638 [56439] (92840} [56 439] (47257) [ (99229 [33429] (63896) [3604] ( ) [78375] (64533) [60026] (6796) [57690] (67945) [7759] (2007) [79929] 9266 ( 7 834) [9266] (27822) [ (23652) [ (06884) [24464 u 85 TJ [. \J u 8 7 o o [ w XJ v 3 t u o v T o o 7 z o 6 [ 2 4 [ u v F xed (X,T} HK $ o o o o 2 Havlle HK $ w t h pob, e s t cton 207

220 Table 9.2 D. Z ' s system In Hong Kong f o 5-yea peod H a v l e F x e d (A s T) ognal pob. pob. est. aest. b HK $ HK $ ognal HK $ HK $ pob. p o b. e s t. a e s t. b HK $ HK $ (378) (3837) (309466) I 8] [7 [8 S (4263) (3700) (368589) [38] [ (272426) (273490) (2597 6) [33] [7] [24] ( ) ( ) ( ) [4403] [6503 [ ] ( ) ( ) ( ) [56] [34] [ (286269) (232805) (74488) [696] [879] [5740] Notes f o T a b l e 9. & 9.2 : () T h e n t a l w e a l t h f o a l l of t h e above c a s e s s HK $ 00,000. () T h e n u m b e s w t h o u t b a c k e t s a e f n a l w e a l t h, t h o s e nsde b a c k e t s a e t h e coespondng t o t a l b e t s a n d t h o s e n s d e s q u a e b a c k e t s a e t h e coespondng v a l u e s of mnmum c a p t a l d u n g t h e bettng hstoy. () I n Hong Kong, dependng on t h e n u m b e of h o s e s In t h e a c e, e t h e p l a c e o show b e t s a v a l a b l e n o t b o t h. F o m T a b l e 9., n most of t h e c a s e s (35 o u t of 45), t h e mnmum c a p t a l u n d e o u f x e d (X,t) model s h g h e t h a n t h e coespondng c a s e of t h e Havlle model. The f n a l w e a l t h b a s e d on t h e f x e d (X9) 208

221 m o d e l Is bette than the coespondng case of the Havlle model fo moe t h a n h a l f o f the cases (29 out o f 45, b u t t h s m a y n o t b e stong enough to conclude that ou model Is bette n the fnal w e a l t h. I n f a c t, n some cases thee a e only a f e w b e t s each y e a, s o w e m a y n e e d m o e a c e s f o c o m p a s o n, a n d thal: I s t h e e a s o n w h y w e a l s o consde t h e e s u l t s n Table 9.2. F o m t h e c u m u l a t v e w e a l t h e s u l t s n T a b l e 9.2, t h e f x e d (X,t) model h a s m u c h h g h e v a l u e s of b o t h t h e f n a l w e a l t h and t h e mnmum captals t h a n t h a t o f t h e H a v l l e m o d e l e g a d l e s s of t h e a - l e v e l a n d t h e l e v e l of p o b a b l t y e s t c t o n. L e t u s look a t t h e e s u l t s n Meadowlands n T a b l e 9.3. I n M e a d o w l a n d s, w e h a v e s m l a e s u l t s t o T a b l e 9.2. O u m o d e l a p p e a s t o m p o v e o v e t h e H a v l l e model a l o t n s h o w b e t c a s e s. As a c o n c l u s o n, o u m o d e l e s u l t s I n l o w e s k a n d p o s s b l y h g h e p o f t s / l e s s losses (n longe t e m ) w h e n compaed t o t h e Havlle model. Although o u m a n s s u e s t o c o m p a e t h e t w o p o b a b l t y m o d e l s, w e c a n a l s o o b s e v e f o m T a b l e s t h a t h g h e a - l e v e l s l e s s s k y (hghe mnmum captal) n geneal. Smlaly f o t h e pobablty estctons. 209

222 Table 9.3 (a) 9 D. Z s s y stem and t s va at o n s n M e a d o w l a n d s f o a = K 2 Model T" o,. o d e ng, pob Case ognal US $ Havl e US$ US $ Place onl y (70883) [7090] 6846 (70673) [79] (69648) [ 7 9] Show onl y 27 (5493) [49 22 (54995) [49] 63 ( 53648) [09] Place Se S h o w 2007 (2525) [ (25887) ( } [32] Place onl y 724 (2238) [ (2585) [873 Show 9976 (436) [6898] 072 (40767) [7025] P a c e Show 7388 (62762) [782] 8470 (6862) [770] u VJ o o 8 o T ) TJ ( 5 [ f v 3 [ o 8 o ( x, t ) w t h pob. ^.. estcton v Fxed wth pob. 丄 estcton

223 Table 9.3 (b) D. Z ' s system and t s vaatons Meadowlands f o a = L6 Model fo odeng pob. Cas ognal US Havlle Fxed c x tf ) P a c < onl y wth pob. e s t c t o n us$ w t h pob. e s t c t o n us 8803 (48046) [976] 9003 (47906) [976] 752 (47228) [976] Show onl y 445 (08604) [432] 450 (08655) [439] 407 (0844) [397] P a c «& Sho 4296 (6525) [3895] 4448 (6733) [405] 3378 (58735) [3008] Place onl y 3472 (2343) [8456] 444 ( ) [ (0547) [876] Show onl y (24555) [988] (24255} [9255] (23658) [9259 Place 8c S h o w 563 (36466) [ (35555) [8464] 4539 (33763) [8388] 2

224 Table 9.3 (c) D. Z s system and t s vaatons n Meadowlands fo Model fo odeng pob. oc =.20 Cas e ognal 2467 (76947 [2430] 2484 ( [ (28097) [347] \J u %J TJ VJ 2 l L o (2204) [8099] Pla c e & Show 22 (3990) [845 [359 ( n Show (8488) [9080] [8883 Place only F x e d (X, t ) ( Place Sc S h o w 6403 (26297) [8883] $ Show only US $ oo s 5o Havlle Place only w t h pob. e s t c t o n s u US $ wth pob. es t c t o n 2859 ( 7 80) [9 330] 0972 (328) [8490] 387 ( 9 980) [8330] Note: ⑴ The n t a l w e a l t h f o a l l o f the above cases a e U S $ 0,000. () The numbes w t h o u t backets a e f n a l wealth, those nsde b a c k e t s a e the coespondng t o t a l bets and those nsde squae b a c k e t s a e t h e coespondng mnmum c a p t a l d u n g t h e b e t t n g hstoy. 22

225 9.4 S p e c a l c o n s d e a t o n n J a p a n In the above secton, the complcated o d e n g pobabltes when nomal d s t b u t o n of u n n n g tmes s assumed. In t h s secton, w e w l l apply the appoxmate method suggested n secton 8.4 o f Chapte eght t o D.Z's system n Japan snce we have empcally shown n Chapte e g h t t h a t t h e Gamma model has a b ette f t n Japan. As we have seen n secton 8.3 t h a t t h e maxmum l k e l h o o d estmate o f the shape paamete () s 8.8). We use t h e f o l l o w n g f o m u l a f o place e t u n on hose p e d o l l a f hoses and j f n s h f s t and second, egadless of t h e ode (l-t)(pl+l) = + INT [ * 0 + / 0 2 (+Pl^ n Japan, t = t a c k t a k e = 26% n J a p a n. (Ths f o m u l a s not qute coect a s t h e e x a c t f o m u l a s not known. I t s only used f o compason puposes hee.) T h e c o m p a s o n s a m o n g H a v l l e, H e n e y a n d S t e n (=4) models b a s e d o n t h e e s u l t s of D. Z ' s system, ( w t h 0:=.6) a e s h o w n n t h e followng table. 23

226 Table 9.4 Compasons a m o n g dffeent models n Japan Fnal Wealth Model 00 yens T o t a bet Mn mum captal 00 yens 00 y ens Hav l e Heney (A=0,76, T=0.62) S t e n (=4) CX G ( 5 ) = 0.88, t G ( 5 ) = 0.8) Fom the above table, the Sten model only has a small mpovement ove the Havlle and Heney model n tems of poft. Also, the d f f eence n mnmum captal between the Sten and the Heney model s small, but both models have hghe mnmum captals (.e. less sky) than the Havlle model. Anyway, we suggest usng ths appoxmaton because the Sten model (=4) has a bette f t to the data, 9.5 Recommendatons Fom the above analyses, we can have the ecommendatons fo the mpovement of D.Z's system : 24 followng

227 9.5. Use fxed X and x model to obtan bette estmates of complcated odeng pobabltes f w e ae wllng to assume the unnng tmes follow ndependent nomal dstbuton^ If G a m m a d s t b u t o n of unnng tmes s assumed, w e m a y use the appoxmate the complcated odeng pobabltes If bettos want to be safe, a pobablty estcton can be added n ode to sceen out the pobabltes of bettng longshot hoses. Zemba and Hausch (987) ted the D.Z s system n many a c e s n U.S. and Canada and t h e y epoted t h a t t h e system woked qut e w e l l n g e n e a l although t h e pobablty estmaton c a n b e mpoved. Thus, f u t h e nvestgaton s equed t o undestand w h y D.Z's s y s t e m cannot poduce p o f t s n Hong Kong. 25

228 CHAPTER TEN OVERALL CONCLUSIONS Although T h a l e a n d Z emb a (988) concluded that ' "modellng gamblng behavou s complcated " we nevetheless have done a lot of analyses on the acetack bettng maket. Fom the esults obtaned n the pevous chaptes, we have the followng man conclusons and comments. 0. Bettng behavou In Chapte thee, we have poposed a smple model fo studyng the effcency of the wn bet maket. The esult s that the favoute-longshot bas does not exst n all the data sets we obtaned, and we fnd no eason to suppot any stong concluson of a genealzed sk pefeence among gambles. All the empcal esults ae consstent wth the pevous studes. 0.2 Complcated bets analyses In Chapte fou, we have appled a logt model to analyse dffeent types of complcated bets. The man esult Is that to pedct the complcated odeng pobabltes fo a patcula bet type, t s easonable to use the bet factons of that bet type. And t s had to beat ths by usng wn bet factons togethe wth cetan models. 26

229 0.3 Models f o pedctng odeng pobabltes F o m Chaptes fou, fve, seven and eght, w e have found that s o m e othe models based on dffeent assumptons of unnng tme dstbutons ae bette than the smple Havlle model. Howeve, by consdeng Gamma and Nomal dstbutons (.e. the Sten and Heney models), we do not have a consstent concluson as to whch dstbutonal assumpton s moe appopate n all acetacks. We have also ftted ou logt models n dff eent acetacks and, n geneal, we have ejected the Havlle model. It s Inteestng to note that ou logt model wth cetan fxed paamete values povdes good appoxmatons to both the Heney and Sten models. Futhe nvestgatons may be equed n ode to undestand why the unnng tme dstbutons ae not so consstent n dff eent acetacks. A genealzed dstbuton (e.g. genealzed Gamma dstbuton whch ncludes Gamma and Webull as specal cases) may be ted to mpove the f t futhe. Howeve, we have to povde good appoxmatons fo moe sophstcated models, othewse, these models cannot be pactcally useful. An altenatve way of analysng the unnng tme dstbutons s based on the exact unnng tme data dectly (If avalable) athe than the outcomes of hoses n tems of the anks. Howeve, then we have to take cae of the possble dependence effects In some aces such as the effect of same dstance and same weathe condtons whch may affect the exact unnng tmes. 27

230 0.4 Bettng Stategy We have consdeed the D.Z system and appled ou methods dscussed n Chaptes seven and eght n dffeent acetacks In Chapte nne. The esults ndcate that thee s an mpovement ove the ognal stategy geneally. Howeve, we do not have confdence that ths stategy wll ean much poft. Seveal easons ae possble. Fst, we have used the wn bet factons alone and not consdeed the othe factos such as the tack condton, weathe condton, ablty of jockeys, etc. Futhe eseach wok s possble n applyng logt model usng othe factos as covaates f they ae avalable (e.g. Bolton 8 c Chapman (986)). Second, though the Kelly cteon has some attactve popetes, fo example. It maxmzes the pobablty of exceedng a gven wealth level In a fxed amount of tme and t maxmzes the expected gowth ate of wealth, the optmal bet amounts may appoach total wealth. Recovey may not be possble. One dea s to consde gowth and safety togethe. MacLean, Zemba 8 c Blazenko (992) popose to use the factonal Kelly cteon n geneal nvestment and gamblng stuatons,.e. Invest n the sky potfolo wth a cetan facton of the optmal amount detemned by the ognal Kelly cteon. Thus gowth can be taded fo geate secuty by nceasng the nvestment facton n sk-fee nstuments- Futhe theoetcal and empcal wok ae equed to popose a good bettng stategy n ths acng aea. The 28

231 last c o m m e n t s that the estmated pobabltes m a y not be accuate enough. Ths addtonal uncetaty In the optmzaton m a y handled b y usng a Bayesan appoach ( B a w a B o w n 29 be Zc Klen (979)).

232 APPENDIX A ESTIMATED PARAMETERS FOR A-CLASS AND L-CLASS MODELS In ths appendx, we epot the estmated paametes f o all A-class and L-class models. Table A. A-class models f o 6 - h o s e - a c e s A, a s 台 s model I o,p a, & a., a , , , , , , , , ,-0,2776 a a a ,-228,-2249,.3557, ,0.0553, , , ,.3585,.3923, 卩.5659, ,. 5 4 山4526,.682,.4978

233 Table A. 2 A-class mode s f o 7-hose-aces : model I(paa ) o,js cc, ,3, a , a 芦 a a a s ,.9,.920,.242,.3200, ,0.3398,0.2589, ,0.274,0, a, , , - 0, 0 5, , , 合 S A, , , , ,.909,.99,.2424,.399, ,. 7 3,, ,.37,.3820,,0663 T a b l e A. 3 A-class models f o 8-hose-aces model 0,p I(paa ) , ,0.2757,0.3463, ,0.270,0.226, , oc, ,.398,.938,.45,.799,.2246, ,-0.904, , , ,0.0455, , ,0.2287,0.0420, 0 002,0.0252,0.359, ,-328,.857,.356,.685, ,.262,.953,.0,.920,.397,.2409

234 Table A,4 L - c l a s s models f o 6 hose-aces : model I(paa.) 0,^ A a»s ,.3505, ,-2609,,930 a' ,0.0553, , , a,冷 X ,-0,3978, , , a ,.64388,-0.279, , 合,S.3366,66,.8432, ,.2056, Ta b e A. 5 L - c l a s s mode s f o 7-hose-aces : model I(paa.) o,p , a, ,0.3398,0.2589, ,0,274, ,-0.070,-0.05, ,0.0694, , , , , ,.2633,.2666,.2237,.960,.2036, ,.869,.4439,.432,-2874,.277,.333

235 Table A.6 L-class m o d e l s f o 8 - h o s e - a c e s : o p , ,.28,.2008,.230,.2025,.80,.933, ,0.2757,0.3463, ,0.270,0.226, ,-0.866, , , - 0, , , /3-2683, , ,-0.849, , ,-0.707, a?, 3 a 台 s a A, C O s I( p a a ) model ,. 28,.072,.702,. 870,.299,.3850 Anothe s e t o f models : (a,s,p,^) a = a + B = 卩 + y and ' ' T a b l e A.7 A - c l a s s models f o 6 - h o s e - a c e s mode I ( paa, e s t. paametes.426, ,0^, na, o ,.3363, a,6,p, , ,.2930, , ,

236 T a b l e A.8 A - c l a s s m o d e l s fo 7 - h o s e - a c e s I Cpaa ) mode est. p a a m e t e s 0,0 /3,y ,-0.D476 oc 6,^, ,.2787,0.05,0 oc, , ,.920, ,. 9,.2420».3200,.2430 T a b l e A.9 A-c a s s m o d e l s f o 8 - h o s e - a c e s mode I (p a a. ) 0, a,6,p, a, H e s t. paametes , ,.244, ,0.0639,.2606,.2389,.2750,.20,.200,.94,.0939 T a b l e A.0 L-class models f o 6 - h o s e - a c e s I (p a a. ) mode e s t. paametes 0,5,, , 0, p, , ,5,^, , ,5,p, - 228, ,.3864, , n, I , , ,.2472,.235, ^ , , , ,.3366,

237 T a b l e A- L - c l a s s models fo 7-hose~ac e s I ( paa. mode est. paametes 0,5,, ,0,l , ,6,0, ,.2798 o,5,p, ,.2526, , ,.2784, , ,.225,.2536 a 0 3 亨 , , , , ,.2957, T a b l e A.2 L - c l a s s models f o 8 - h o s e - a c e s I (paa.) mode e s t. paametes 0,5,, , 0,J3,了 , ,6,0, ,.2448 o, H , , , 6, 0, ,.3 72,.2567,.523,.635, 32, ,. 0 2,.072 a t,0 ^, T , , 0, , , ,.2709,

238 APPENDIX B G A U S S I A N - H E R M I T E I N T E G R A T I O N S F O R C O M P L I C A T E D PROBABILITIES Ths appendx gves the fomulas of Gaussan-Hemte tfecta jpofoatdltes. These f omulas ntegatons fo exacta and have been used fo the analyses n Chapte fou. Unde Heney model, T = the unnng tme of hose u = P(T j < T j N(0 ) ndependently. < m n {T >) < ^ ( u j 去 j ) I T [ $ ( u ^ ( u ) d u, j J h +0-0 ) J j ^ J [ - $ ( / 2 z +0 - e ) h j and. jk P(T < T j < T < mn k, j,k {T >) u+0-0 $ ( v ) <^(v) d v ] j K [一$(u+0-0 )]泠 u) d u ^jk k Vzx +0-0 yu w -J n h VTT ^ k [-^ (Vzz +9 -e ) [ h k 226 h k j ^(v+e - e ) <t>v) dv] j

239 whch and ae weghts and oots suppled b y c o m m o n l y used table fo Gaussan-Hemte ntegaton (e.g. A b a m o w ^ z and W e have chosen 2 0 pas of weghts and oots In ou analyses. T h e Intenal ntegal of jk 227

240 APPENDIX C DER I V A T I ON O F A P P R O X I M A T I O N F O R PROBABILITIES U N D E R EXTENDED M O D E L Unde the extended model fo vaance and coelaton stuctue. U d whee U, U, U ndep. e. T ^ u - o N( =,2, ^ N(0,)» 0^+ A ' u0c» ), and t h u s the f o m o f t h e c o n d t o n a pdf s : y-0 ~ 旧 f (y u ; 0 ) = e x p { [- 4 u c c. whee \fj Y = { + e x p [f+<(0.- 0)} and = e x p [ b ( 0 厂 0) n =P(T< m n {T }) o = ( g # o U " F ^ y l U o ; 0 ^ f ( y l U 0 ; e )d y )> n d g (0) ^( ) + j= j 0 = o It s s t a g h t f o w a d t o show t h a t g (0) = $ ^ l / n ) > (c.3) (C.4) 2 )

241 The second tem nvolves (0) a e <f>x 0=0 o) 3 e f (y U j dy A whee l o;e ) 0=0 Cl/n). To d e v e a smple appoxmaton, w e assume t h a t, f o 萁. d\j t ^ oq d dc 0 and od ^ 0 and thus, 0=0 When = j, (z (0) 0=0 f J["F(ylUo;0) 州 V V ae dy 少(u ) du (C.5) 一 E ( y 0=0) y - u /(l+e f ) Let v - / (l+ef }2 V a ( y 0 = 0 ) Gven t h a t ln^(x) s quadatc n x t s e a s l y shown t h a t a Inf ( y l W 0=0 A, + B'tu ) z + CTz 229 (C.6) 0=0

242 whee A = - b f f 2 - ) +u [ b + ( b - d ) e f ] / f! B (u ) = _ - / f f = + e and = - l/n Thus, (C.5) becomes d g (0) 沴(zj 0 I ^0=0 [l--$(z)]n"^(z) [A, + B ' fu z + c ' z 2 ] d z ^(u ) du [A, + B'Cu ) n whee a u ;n (2) + cuo = t h e x p e c t e d s t a n d a d nomal ode s t a t s t c,. = t h s e c o n d moment about o g n o f t h e standad nomal ode s t a t s t c, [A, + whee + C ( 2 )] CC.7) B = (-/f, 2 ) " / 2 230

243 When 垆j, d ge) 0 沴(Zq)丁 j '0=0 j 0 = --00J J J 00 y 6 n f ( t I u ;0 ) ] vv [ ae ^ -0Q -00 f ^ y l W - [ ^ ( 2 ) t j m丨v9?dtl dy ^(uo) duo I ( A, + 0, ( ) v + C ' v ) ^ ( v ) d v ] (j>(x) d z ^ ( u ) du 0 0 0,00 by (C.6) y-u/ f ' whee (-/f, 2 / 2 ) t-u/ f and f ' = +e (-/f ^ 00 ^ - } [ dz - [ - 0 (z ) J J n co L ( n - ) ) / 2 (A,+33,( )v+c9v ) ^ ( v ) d v ] d z ^ ( u ) du [A,+33,( J z + C ' z 2 ] ^ ( z ) d z ^ ( u ) du u s n g n t e g a t o n by pats " J : [A, + ( [ A, n(n-l) 2 + B fx + C, /! : 2 ) l;n n 23 2 ) J du (C.8)

244 S u b s t t u t n g ( C. 4)» (C.7) a n d (C.8) nto (C.3), w e have (0 «(l/n) by assumng 7 ~ = 4 ^ 7 ~ t [ A + B + C f(2)] 严 l;n ( n - l)0(z ) E 0 = =0 j= j Theef oe, f u 0(z ) (n- DCz! whee z =$ (C.9) XP), (l/n), P = wn b e t f a c t o n f o hose I, M = A B^fx +C9[2), I ;n n fj. = t h expected s tandad nomal ode stat s t c. (2) t h second moment about ogn of standad nomal ode s t a t s t c Now, we ae gong to fnd a geneal appoxmate fomula fo (m=l,...,n)- Fo smplcty, we defne = P( T TT * = / = m hee. <... <T < mn { T }) I m ^,......,m P( T <.<T m <A <...<A ) m+ n = H h(0) > m+ w h e e A,...,A s some pemutaton of m+ n T»m+ summaton s taken ove all possble pemutatons. 232 T n and t h e

245 and each tem n the above summaton s P( T <".<T I <A <...<A m m+ ) n 0(U ) whee J f 2 a n-x (t u ] 2' 0 f (a I u ) da... dt n n 0 n l (a Iu ) ", f (a lu ) m+ m+ 0 n n 0 ae du 0 the pdf of A m+ espectvely. Usng f s t ode Taylo s e e s appoxmaton agan. d h(0} h(0) «h(0) + J] 0=0 (.0) It s e a s y t o show t h a t h ( 0 ) = 否 < (C.ll) n(n-l).(n-m+l). Let a = h(0) In (C.ll). whee When s = l, h(0) 0=0 a e m+ f ct l u ) l 0 u da n... dt 233 l ] ^ ^ 0=0 f ( ct l u ) ) du 0 C

246 3nf m+ 丨 l (t Iu ) l! 0, f ae u Let f ( t u l V d a /(+e) n 0 = 0 ^ fo = m+. - l/(l+ef)2 Recall (C.6) t h a t : d Inf (t Iu ) ~ d a w h e e A', 0 = A' + B (u z + C z 2 0=0 o Q n and C, a e d e f n e d pevously, hence, a hfe) ^ - J - f - 0=0 [A'+Blu ) z + ) (2 m+ <f>(z ) dz dz (n ""7)! f n [A, l;n.., dz I ^ ( u ) du C /(2)]少(u ) du l;n o,and fo - l/d+e )' /(+e du

247 Smlaly, fo ( ^ )J ^ we have : a h(0)丨 ^ ^ = M / P d Q 通一一 0 = 0 n m (C.2) whee f 2) M = ;n ;n When s > m, d he) 沴(a "a~a s e=o 補» P( T <.<T <A <."<A m m+ n 9=0 <...<A P ( T <.. <! <T <A m+2 n l l m+2 P( T <-"<T\ <A = ( n - m ;) n! (M <T ) M ) m+ n ( C. 3) = V M / P u n m+ =m+l 235

248 Substtutng (C.ll), (C.2) and (C.3) nto (C.0), w e fnally obtan n (C.4) because : n I M ^ =l n = na + B, n = ;n (2) + C, E fx = ; n n (2) = n A, - A' T] j = n $(y)~[l-$(y)]n""0(y) dy We may scale (C.4) so that the sum-to-one constant y xu L l 5,...,5 = s satsfed. 236

249 APPENDIX D DERIVATIONS O F S E C O N D - O R D E R APPROXIMATIONS FOR THE HENERY M O D E L Fst, we have to compute 0 by usng the bet factons P Then w e can u s e t h e 0 t o compute n fs. We e x t e n d t h e second appoxmaton J d e a n Heney (98) usng second-ode Taylo s e e s expanson n ode t o have moe accuate esults. The d e t a l s a e a s f o l l o w s. We f s t gve some u s e f u l fomulas. (2) Defne u and ll b e t h e f s t Sc second moment o f t h e t h standad M;n ^;n nomal ode s t a t s t c n a sample o f s z e n. Ll, (2) o jl u^(u)2[l-^(u)]n""2 du = n ( n _ h n Poof LHS = - T 口-巧) u6cu) n- [l-$ (u)] n " [ 0 ( u ) - u 0(u d u RHS 237

250 Poof LHS = 一 ~ J 0(u) 0 [l-^(u)]n du RHS.3 o 2 _ 2 $(u) [ 韻 广 3 du =n ( n H Poof LHS= - - 巧 u ^ ( u ) $ ( u ) n-z n- [ l - $ ( ) ] n " [ - u 0(u)$(u) + 0(u)$(u) + u^(u) ] du 00 t(2) f(2) ~ - ( ~ n n(n-lj RHS ] by.

251 Poof :l-<l>(u)] n-2 LHS (pu^u) [l-^(u)]n"2[-u0(u)^(u} + ()2] du n- by RHS.2.5 2(-/(2)) «J 伽 ) [ d - _ -00 =U - 制 二 Poof LHS= - [ 广 W 丨 2 ^ -oo Ky f n 2 " J 一 [ l - $ ( u ) ] n " 2 2^(u) I-u0(u)] du O C =RHS by. L6 3 J ) ㈨ 广 ㈨ d u =- o 0 (2) (2). nd^du^jcn^) -00 Poof LHS : + n-3-2 ^(u) $(u) -oo.00 [ K u ) ] n - 3 { 20() [-u^(u)] ^(u) + ()3} du 239

252 «fn (2) (2). 2 ( 2 - / - f ) ; n «2 :n (n- (n-2) (n-3) 2 ( - (2) n n(n-l) (n-2)(n-3) b y L 3 and L 5 RHS Now, w e can poceed t o t h e f s t stage,. e. second ode Taylo s e e s f o computng 0. Let T b e t h e unnng t m e o f hose, 了广 NCe^l) ndependently and s the pdf of 丁 厂 Let hc0) = ^- [ P(T < o t h e s ) ] n [ - F ( t e ) f ( t e ) dt I ) I [ - $ (u+0-0 ) ] () du ah JW h(0) ae 0=0 [e: d h I I e j ek f ae ae j^k 0=0 (D.l) Let z =^^(/). o It s easyj t o show t h a t h(0) =z. o Fo t h e second pat o f (D.l), 240

253 W h e n j萁 ah ae = H. [l-^(*-0-0 )] () 0(+0 ~0 ) du JUUFH j dh JW [l~^(u) n»0 = 0 by.2 ^(z ) n ( n - l ) When ah at" 一 = ^(h(0)) n [-F (te )] f (te ) (t-e ) dt ah [一 $ ( u ) ] n u0(u) du (pz Fo the thd pat of (D.l), When j^, ae»0=o ) n 沴(zj 萁 j [-^(u+0 - e )]々 ) (u+0 - a ) j 沴(u+0-0 ) du [-h(e) ^(h(0)) 24 do [l"-$(u)] n 2<f>(u)2 du e=o

254 When j =, ae ) (pu n [-F ( t 0 )] [f (tl 0 ) (t-e ): f ( t 0 ^ ] dt ah 0(0)) 卜 h ⑷ ae" [l-$(u) ] n " (f>{u)u d u : 2 ) n tn-l) D Hz) F o the last p a t o f (D.l), When j ^ and k *. azh ae ae j 0=0 沴 z j ^[ ^(+0^ 0 )] ^(u+0^ 0^) (+0厂0 沴 c () du 242 0=0

255 o - [ - h ( e ) 0Ch(0)} = k ] -oo o f c 2)> -2( - fl h ( n ^(u) \l du j 0=^ Z [ n ( n - l ) ( n - 2 ) 叫 + o( { 2 IuSt ) } ^ When j= and k *. a ae ae»0=0 袋仏[-$(u+0 ^-0 ) ] ^(u+9^-0^) () d u 沴 ( [ - h ( 0 ) 0(h(0)) ae I [l-<>(u)] n "VC u )u d u 0=0 (2) n(n-l) t c ^气 ) ^(zj' Substtute a l l t h e p a t a l devatves n t o (D.l), we have P(T < othes) 沴(zj(n-l) ( a j e a 2e 垆 j J ^ fo =,2,,!! 243 e e j 关k j,k搿 ^ 界 k^l by L l

256 whee A,A 2 j B and a e defned above. Note that the f s t two tems n the above backet ae the same as those n Heney's f s t ode appoxmaton fomula. Now, we can poceed to the second stage,.e. appoxmatng usng second ode Taylo sees expanson. g(e) = $ [P(T < T < o t h e s ) ] [-F (te )] F (te ) f ( t e d t j j [-$ (u+0-0 ] (+0-0 ) V " -oo g( ) + ^ k0 f If" *0=0 E e ae ae 50 0=0 (D.2) L e t a =少 L n(n-l)j It s easy to show th at g(0)=a. F o the second p a t of (D.2}, When k萁 and k气j. 0 dg^ f [ $(u+0-9 )] $(u+0 0 <p{\+q 0) W 一 (pge))j ^ jk j j j k (pn) du 244

257 When k=, Sj["$(u+00)]伽+W _ ae = ^ ( g ( 0 ) ) J dg I ^ a 0 = 0 一# n ( n n d u 一 by.2 ] ) When. k = j, If - jum I j «a F 0=0 dt - 03 ""^ a ) n ( n - l ) F o t h e t h d p a t o f (D.2), W h e n k关 and k气j, 0(g(a))2-^ ae e=o k / 00 = 0(g(e)) u^(u)2 $(u) [l-$(u ]n一3 du V 03 - [ - g ( 0 ) ^Cg(e)) - ] 麵 00 T ^Ik [l-^(u+0-0 )] ^(u+0-0 ) ^(u+0-0 ) J j j k

258 When k=. 於(g(a))2 d g 30 0=0 00 = j 0(g(0)) - u 0 ( u ) 2 [l-^(u)] n "" 2 d u [ 一 g ( 0 ) Oo 一 ;n (a) ^ 一 n(n-l) V ;n a J \, T t b y L J When k=j, 30 广 0 = 0 f o I ^lj =\ ^(a) f 5 [ - F (te )] F f tl e ) t f f tl e ) (t-e )2 00 j j j f (tle^ ] d t - [ - g ( 0 ) ^ ( g t e ) ) " ] - 讀 I JU[-F(tle)] Ft\e) ( ^ 0?d t } j

259 F o the last p a t of (D.2)f When k^ and k关j, () 关 and l^j, ae ae e=o 0(u+0 0 ) T I [l-^(u+0 一 9 ) ] 公 ( Y j k ^jkl j j 0(g(0)) $(u+e - 0 ) ^(u) d u -g(0) 0 ( g ⑷ w <j>(u)2 [l-<hu)] $(u) }l 0=0 沴(a) I ^ 飞 ~ 2 " [ 3 n7 ( n - l ) ( nn- 2U) ( n - 3 ) ( 丄 / a ^ U +U \2 n 2 ; n I n ( n - l ) (n 2) J () l=, 丄 2 a2g 沴⑷ae aek 0=0 n I⑵-2)l \ ;n 2;n J b y.4 a n d.6

260 0(g(Q)) - J ^(u+0-0 ) j TI [-^(+0-0 )] [-0(u+0-8 )] k jk j j () d u -g(0) 0(g(0))5 -] Oo I <(u+0 0 ), J T I [ -^(+0-9 ] $(u+0-8 ) <p(u) d u k ^ljk J J 0=0 ( 2 ). 2 ( - f ll </>(a) n ( n - l ) (n-2) a+ a/ n(n-l)(n-2) f n(n-l) \ b y.4 a n d.5 () l = j. We m a y w t e 0 ae = ^ 丄 f j l 0(g(0)) J ^ljk -00 k [-F (te )] F f t j e l f (tje )f (t e) dt (t = U + 0 ) Then, ae \ ae k»l0 = n0 W ) ) [ J M- IJj k - F ( t e ) ] F t t l e ^ f k (t 0 k ) f ( t \ Q ^ (t-o^) d t [-g( ) ^ g ( 0 ) ) " 248

261 () 2 [ l-<j>(u)]n 3否(u) d u \ I o (2) n 2 丄f (2) p. n 2 n n(n-l)(n-2) 少 ⑷ {- n(n-l) (n-2) f n(n-l) y.3 and.5 When k=, () 萁 and l^j. 30 0=0 [ 0Cu+0 0 )] <p(xo d u j [-g(0) ^(g(0)) I f - ] J ( (U) 2 [l-$(u 广 2 du } ^a, (2 ) 2 ( - Lt ) n / a n(n-l)(n-2) \ U + \ f n(n-l)(n-2) J n(n-l) b y L 2 and.5 () l=j. Wte : 生 一 W _ 0(g(0)). n [-F (te ] f (te ) f Ct 0 ) dt *j 249 j J 0=0

262 Then, ^a)2a/af" 0 j a0= 0 4>(gQ)) n [-F ( t e )](-f ( t e ) ) f f t l e ) ( t - e ) d t ^lj I J j j v. -oo f = - [ - g ( ) ^(g(0)) _ «j -oo / J I 0=0 - \ X f⑵ f / = When k=j, () 式 and l^j,, a e a e I ^ 伞 j»0=0 一 aa ae j l 0=0 () l =, 2.2 a g I ^ d T Wj 0=0 n M M M I I = 2,f s g ^ d T j W H e n c e, (D.2) b e c o m e s : 250 a 0==0 n _ -

263 tt. = P (T < T < oth e s ) (e + a )(烊 J 2;n 0=0 I E 0 l^k aea e 0=0 whee ^ ll- L ( n - l ) L(n-l)0(a) =0t2 [ 0(a) I + a I 2 } + 2 0, 2 [沴(a) I j E ek2 [ _ k^j +a I I5 + 2 ] + ai ], 25 3

264 E 0 e k [ 0Ca) I 7 + a I 6 2 ] E + 萁j k ^ l j k^l 2 0 J L 0 k关 J 2 0 k euk [ 0(a) I 3 Is ⑷ +a I + I ] a I6 I4] + k^ j [ 0(2) I J +3 I 4 I 2 3 > whee (2 ) n(n-) (2 ) 2 n 2 _n(n-l)' - 4 一n(n-l), n(n-l) (2) (2) ^ ; n ( n - ) ( n - 2 ) ' - 2 (3-2) ( 2 ) (2), ( n - ) ( n - 2 ) (n-3}, n ( n - l ) (n-2} 2 (l-r (2) n 8 " n ( n - l ) ( n - 2 )

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