Poděkování. Čestné prohlášení

Transcription

1 Poděkováí Ráda byh oděkovala vedouímu mé dlomové ráe do. RNDr. Dael Hlubka za jeho omo odoru a ovzbuzeí. Mým rodčům za odoru a olk ořebou movaí. A Bohu kerý o vše umožl. Čeé rohlášeí Prohlašuj že jem vou dlomovou rá aala amoaě a výhradě oužím ovaýh rameů. Souhlaím e zaůjčováím ráe. V Praze de 9. Proe Sally Abdel-Makoud

2 Akowledgeme I would lke o akowledge ad eed my hearfel graude o my uervor do. RNDr. Dael Hlubka for h kd hel eourageme ad uor. Mo eal hak o my are for uor ad muh eeded movao. Ad o God who made all hg oble. Saeme of Hoey I hereby delare ha I have wre my dloma he deedely ad eluvely by ug he quoed oure. I agree wh ledg of he he. Prague Deember 9 Sally Abdel-Makoud

3 oe Akowledgme... Saeme of Hoey... Abra... 6 Iroduo... 7 Eoeal famly of drbuo Ba defo ad oe urved eoeal famly.... Mome ad umula geerag fuo..... Seal ae of eoeal famle ad ome of her mome... 7 Suffe a.... Ba oe.... Fuo of a uffe a.... omlee a.... Allary aroduo... 6 Uformly mo owerful eeferee Radomzao e No-radomzao e Fudameal lemma of Nay-ma ad Pearo Drbuo wh moooe lkelhood rao....5 Oe-arameer eoeal famly....6 Veor arameer eoeal famly....7 Ubaede for hyohe eg Lea favorable drbuo... he urvaure eoeal famle he geeral defo of aal urvaure he geomeral urvaure he roere of he urvaure... 8 Referee

4 Abra Název ráe: Eoeálí řídy a jejh výzam ro akou fere Auor: Sally Abdel-Makoud Kaedra: Kaedra ravděodobo a maemaké aky Vedouí dlomové ráe: do. RNDr. Dael Hlubka Ph.D. e-mal vedouího: [email protected] Abrak: ao dlomová ráe vyhodouje Eoeálího řídy rozděleíkeré mají eálí oaveí v maemaké ae. Dlomaka e ezámí e základím ojmy a faky ouvejíí rozděleím eoeálího yu. Seálě e ak zaměří a výhodo eoeálíh říd v klaké aramerké ae edy v eor odhadu a v eováí hyoéz. Důraz bude klade a jedoaramerké víearamerké yémy. Klíčová lova: Saká feree eoeálí řídy ey hyoézy. le: Eoeal famle aal feree Auhor: Sally Abdel-Makoud Dearme: Dearme of Probably ad Mahemaal Sa Suervor: do. RNDr. Dael Hlubka Ph.D. Suervor' e-mal addre: [email protected] Abra: h dloma he rovde a evaluao of Eoeal famle of drbuo whh ha a eal oo mahemaal a. Dloma wll lear he ba oe ad fa aoaed wh he drbuo of eoeal ye. Eeally wh foug o he advaage of eoeal famle laal aramer a hu heory of emao ad hyohe eg. Emha wll be laed o oe-arameer ad mul-arameer yem. Keyword: Saal feree eoeal famle hyohee eg. 6

5 Iroduo Sa ee a mahemaal aroah of makg effeve ue of umeral daa erag o grou of dvdual or hg o ha udyg hee daa how a be olleed aalyzed erreed ad reeed order o awer he queo relaed o he world aroud u. heory of a ake he real daa arg from he raal uao ad ue hee daa o valdae a ef model o make good guee or emae of he umeral value of releva arameer or eve o orgae a model. Sa uually dvded o derve ad fereal a. Geerally derve a derbe he olleed daa ad how how look lke. Derve a lude frequey ou rage hgh ad low ore of value mea mode meda ore ad adard devao. Varable ad drbuo are wo eeal oe o uderadg he derve a. Ifereal a are ued o make geeralzao or redo baed o he dero of daa. Oe key ba oe of fereal a aal eg. Saal feree aal duo ad fereal a are he roee of drawg oluo from he obervao of daa ha are ubje o radom varao for eamle obervaoal error or amlg varao whh a o be avoded. here are wo ye of aal feree: emao of oulao arameer ad hyohe eg. Saal model oder a robably ae E E P θwhere P θ a drbuo o meaurable mer ae EE uh ha E aroraely hoe ae whh all oree meaureme a be oberved ad E a gve σ-algebra o ha E oa all he ube of Ε ha are releva o he roblem we aume ha eah obervao E he realzao of a radom varable Χ wh value Ε ha defed o ome uderlyg robably ae Ω F P where P a robably meaure o Ω F. Suh a radom varable a mag Χ: Ω F E E ha E - F meaurable. o defe he aal model we aume ha a famly of robably meaure {Ρ θ θ Θ} defed o Ω F uh ha for every θ Θ he drbuo of Χ uder Ρ θ gve by Α = P θ Χ Α Α E. ombg he amle ae wh he e of oble drbuo of Χ he aal model a be formulaed a M= E E P θ θ Θ. 7

6 Saal deo roblem o be able o oru he roblem we eed fr o hooe a uable famly {Ρ θ θ Θ} of drbuo of Χ o E E where Ρ θ rue bu a lea arly ukow drbuo. he arameer θ aumed o be kow oly ha le a era e Θ he arameer ae. Ad he aal feree ug hee ool o geg formao abou he drbuo of Χor he arameer θ. Mahemaally he roblem hould be formalzed order o make more ree oluo. Le D be a hoe o-emy e whh refer o e of oble deo ad all he deo ae. Our ak o fd a rule whh ell u whh ao o ake for eah e of value of obervao uh a rule a fuo δ E D o ha δ= he deo whh made afer E ha bee oberved. Now how δ hould be hoe. A he begg we odered ha he obervao of daa are ubje o radom varao ha a o be avoded uh daa a o lead o he be deo. Aordg o ha here a be a lo owg o deo; h lo a radom varable ha a be ereed a θ d: Θ D R. Se we are o ure abou deo here a be a rk hu we are lookg for he deo δ ha lead o he mmum rk for every θ Θ. he rk of a deo a be formulaed a R L L dp If we ould make a ef aumo abou he la {Ρ θ θ Θ} where he drbuo of Χ uoed o be he ruure of he ae Dof oble deo d ad fally he form of he lo fuo L he we a ge he oree ouome for our roblem. Emao ad eg Emao of arameer eg of hyohee eleo of oulao ad lafao are eal ye of deo roblem. o emae he value of θ we eed o reae a a whh odered a a fuo of he obervao ad doe o deed o θ wh ome arorae roere uh a he amle mea ad he amle varae. If he a whh ha hoe loe o he oulao arameerθ he odered a a emaor of θ. For eamle he amle mea a emaor for he oulao mea. Now he queo f we aume ha we kow he amle mea ad he amle varae; how a we make ure ha he radom amle ha o oher formao abou he oulao? he awer wll be aordg o whh famly of drbuo we hoe o derbe he oulao. h dea lead u o he oe of uffe a. More deal wll be dued he fr haer. 8

7 haer Eoeal famly of drbuo. Ba defo ad oe I robably ad a a eoeal famly or Kooma-Darmo drbuo a la of robably drbuo harg a era form wll be efed laer. I ad ha uh drbuo belog o he eoeal la of dey fuo. Eoeal famle lude everal mora drbuo ad alo have everal roere. Mo of aramer famle of drbuo a aal model are eeal ae of eoeal famle uh a he ormal eoeal gamma bea bomal mulomal geomer egave bomal ad he Poo famle. hey have he ame roere a a eoeal famly. he famly of drbuo {Ρ θ θ Θ} uh ha wh ree o σ-fe meaure μ we defe d he dee a d For every e h d. A Ad alled a eoeal famly where ad are he real-valued fuo of he arameer ad h o-egave meaurable fuo. he value θ alled he arameer of he famly ad he value ofe a veor of meaureme whh ae a meaurable fuo from he ae of oble value of o he real umber. he eoeal famly ad o be aoal form f for all. By defg a raformed arameer alway oble o over a eoeal famly o aoal form ad rewre a D e h. 9

8 he aoal form o-uque e a be mulled by ay o-zero oa a he ame me relaed by /. he fuo equao. o-egave ad a robably dey fuo f ad oly f D e h d. A oa D > e f e h d he e of o H {.... : e h d } alled he aural arameer ae of he famly.; alled he aural arameer ad he Euldea alar rodu of he veor... ad... heorem.. he aural arameer ae H ove Proof Le we have e H h d e h d e e h h d h d e e h d H Eamle. Normal famly If ha he N Lebegue meaure drbuo he θ= N ad he dey wh ree o

9 e A wo-arameer eoeal famly wh eoeal famly wh aural arameer = ad he aural arameer ae R. h D l Defo. Θ ad o be udefable o he ba of f here e for whh P P. Eamle. Mulomal I deede ral wh + oble ouome le he robably of he h ouome be eah ral. If deoe he umber of ral reulg ouome... where eah ral reul ealy oe of ome fed fe umber of oble ouome wh robable ad he he mulomal drbuo M... ; he he jo drbuo of P!!...! h a be rewre a e log... log h Se he add u o h a be redued o log log /... log / h e

10 h a -dmeoal eoeal famly wh log / D log log e he aural arameer ae he e of all... wh. urved eoeal famly A veor eoeal famly ad o be urved f he dmeo of he arameer veor... l le ha he dmeo of he umber of fuo of he arameer veor... m I Eamle. mgh be he ae ha he mea ad he varae are relaed. I uh ae whe he aural arameer of he drbuo are relaed a o-lear way we ay ha. or. form a urved eoeal famly. Eamle. urved ormal famly For he ormal famly of Eamle. aume ha υ=σ o ha e υ>.7 However h a wo-parameer eoeal famly wh aural arameer h arameer geeraed by he gle arameer υ. he wo-dmeoal arameer le o a urve makg.7 a urved eoeal famly. Eamle.5 Normal amle Le = be deede deally drbued aordg o N. he he jo dey of... wh ree o Lebegue meaure

11 e A he ae of Eamle. h a wo-arameer eoeal famly wh aural arameer. Eamle.6 Log model Le be deede b... m o ha her jo drbuo P m... m m. h a be wre a m m e{ log }.8 A m-dmeoal eoeal famly wh aural arameer log... m. he quay log / kow a he log of. If he ' afy z... m For kow ovarae.9 beome a urved eoeal famly. z he model oly oa he wo arameer ad.8

12 . Mome ad umula geerag fuo heorem.7 For ay egrable fuo ƒ ad ay η he eror of H he egral f e h d I ouou ad ha dervave of all order wh ree o he η ad hee a be obaed by dffereag uder he egral g. Normalzao of he drbuo Se df D e h d. I follow ha e D e h d allg D he log-aro fuo Now dffereae he dey. wh ree o k D. k k lead o Dffereag ervou equao wh ree o l lead o ov k l D. k l I a eoeal famly he a... arry all he formao abou η or θ oaed o he daa o ha all aal feree oerg hee arameer wll be baed o. For h reao we wll be ereed alulag o oly he fr wo mome gve by. ad. bu alo ome of he hgher mome v v. Ad eral mome

13 {... }..... Ug he mome geerag fuo make he alulao muh eaer uh ha M u u u e u.5 If M e ome eghborhood u of he orghe all mome... ere. M ad are he oeffe he eao of he umula of M a a ower u... u... /!...!... u.6... u umula geerag fuo a be deermed by K log... u u M u u k /!...!... u... u From he umula he mome a be deermed by formal omaro of he wo ower ere. For eoeal famly he mome ad umula geerag fuo a be ereed raher mly a follow. heorem.8 If drbued wh dey D e h he for ay η he eror of H he mome ad umula geerag fuo u ad u of he e ome eghborhood of he org ad are reevely gve by u u A u e e h d e A u A.8 5

14 K u D u D.9 he alulao of mome beome eay whe hey a be rereeed a he um of deede erm. a Suoe... where are deede wh mome ad umula geerag fuo M u ad K u reevely. he M Ad herefore K u... u e M u... u M K u From he defo of umula he follow ha k k r. where k r he rh umula of. b h uao alo very mle for low eral mome. If Var ad he are deede oe fd roblem.7 Var j. j. For he ae f deal omoe wh h redue o Var.. 6

15 .. Seal ae of eoeal famle ad ome of her mome Eamle.9 Skew-log drbuo oder a real valued radom varable wh dey e e Parameer he dey a be rewre a e e e log e log h a eoeal famly wh aoal arameer log e D log log.. log e D log.. log D Var Var e..5 Eamle. Bomal mome Le have he bomal drbuo B o ha for = P q ; q..6 h he eal ae of he mulomal drbuo.6 wh. he robably.6 a be rewre a 7

16 e log / q log q Whh defe a eoeal famly wh μ beg oug meaure over he o = Ad wh log / q D log e From 8 lead o M u q e.8.7 o oba he eeao ad fr hree eral mome of aume ha are he umber of uee Beroull ral wh ue robably ad hee ha where oe or zero a he h ral or o a ue. From. ad he mome of we fd roblem.8.9 q q Var q q q 6q. Eamle. Poo mome A radom varable ha he Poo drbuo P P e...;! Rewrg a a eoeal famly aoal form e! We fd log log D e.. f Ad hee 8

17 K u u e M u e e u. So ha k r for all r. he eeao ad he fr hree eral mome are gve by. Var. 9

18 haer Suffe a. Ba oe Oe of ommo hkg heory of emao oder ha he amle oa formao ha ueful for emag he arameer ad formao ha uele for emag he arameer reag a a wll hrow away oly ome of he uele formao whle omleely reag he ueful formao abou he arameer he a uh ae alled a uffe a for he arameer. A a uffe for a aal model f a formave a he full daa meag ha he a oa all of he formao abou he arameer of he model ha avalable he ere daa radom amle. For ae f we kow ha a drbuo ormal wh varae bu ha a ukow mea he amle average a uffe a for he mea. he amle mea Normal drbuo N uffe for he drbuo mea. Defo. A a ad o be uffe for uderlyg arameer f he odoal robably drbuo of gve P P Or P P. for all deede of.e. Eamle. Poo uffe a

19 Le be deede radom amle of ze from he Poo drbuo. Show ha a uffe a. he jo drbuo of he radom amle f ; e!! Ad he jo robably of ad f ; e Where!! Oherwe Moreover we kow ha he um arameer. m ;! e ha he Poo drbuo wh he he odoal drbuo of he amle gve f ; f ; m \ e! e!! Whh doe deed o arameer a uffe a.

20 By ug he defo above arorae o omue he odoal drbuo order o verfy f he a uffe or o fd a uffe a. here e a eeary ad uffe odo for uffey of a a ha ue he uodoal drbuo of he amle ad he drbuo of he a rovded by he followg heory. heorem. Faorzao rero Le... be a radom amle wh jo dey f... ; wh ree o σ- fe meaure. a eeary ad uffe odo for a a o be uffe for ha he dey a be faored a f... ; g... h... Where g a o-egave fuo ha deed o he arameer ad o he obervao oly hrough he a ad h a o-egave fuo doe deed o. he overe rue: f he a uffe he he amle drbuo a be faored a above. Eamle. Normal drbuo Le... be a radom amle from a ormal drbuo N uh ha S Ad le f be he jo dey of... he

21 } e{... f } e{ Now. hu } e{... f.... h g S I uffe for.. Fuo of a uffe a A uffe a by a oe-o-oe fuo uffe bu a uffe a by a o oeo-oe fuo o uffe.e. If uffe ad HS he S alo uffe. Kowledge of S mle kowledge of. moreover ha rovde a greaer reduo of he daa ha S ule H : whh ae ad S are equvale. Defo.5 Mmal uffe a

22 A mmal uffe a defed a a uffe a ha a fuo of ay oher uffe a. A odo for a a S o be mmal uffe S uffe. If uffe he here e fuo f uh ha S f. heorem.6 Lehma-Sheffe for mmal uffe a Le... be a radom amle ad he e of all oble ouome for he amle. he robably dey fuo f of afe f... For ay wo amle o y oder he rao h y f f y Le : eleo be a meaurable rereeao whoe vere allow meaurable Aume ha h y deede of y he... a mmal uffe a for.. omlee a Suoe ha U h a a he U omlee f { E w U } P w U } Fdg uffe mmal uffe ad omlee uffe a ofe mle for K-arameer regular eoeal famly f he famly gve by equao. he kp-ref.

23 heorem.7 uffey mmal uffey ad omleee of eoeal famly Suoe ha... are..d from a eoeal famly e h h If he aramer ae oa a k -dmeoal erval he... where... j k je radom amle ad j omlee mmal uffe a for. heorem.9 Bahadur heorem A fe dmeoal omlee uffe a alo mmal uffe. Eamle. he uffe a from a REF doe o have o be omlee Le be a radom varable form N wh. h famly a REF wh a omlee mmal uffe a. he daa alo a uffe a bu o a fuo of.hee o mmal uffe ad by Bahadur heorem o omlee. O he oher had E bu P o o omlee. heorem. Le he a be a fe famly wh dee... k I mmal uffe. orollary.... k all havg he ame uor. he 5

24 Uder he aumo of heorem. a eeary ad uffe odo for a a U o be a uffe ha for ay fed ad U. he rao / a fuo oly of Lemma. If a famly of drbuo wh ommo uor ad uffe for ad uffe for mmal uffe for.. Allary a ad f mmal Suoe ha V r f he drbuo of V doe o deed o he V alled a allary a for. heorem.8 Bau heorem Le be a famly of drbuo o a meaurable ae. he f omlee uffe a for ad V allary o he deede of V. 6

25 haer Uformly mo owerful e. Radomzao e Le be a meaurable fuo uh ha : ad alled he ral fuo For ay value robably. he hyohe wll be rejeed wh robably ad be aeed wh If he drbuo of P ad he ral fuo ued he robably of rejeo E d alled he ower fuo ad marked by. No-radomzao e I h ae ake oly he value ad uh ha. We are lookg for o ha for H everhele mamum for all K. orreodg o he wo kd of lo fuo he wo omoe of lo fuo a be formulaed a followg: d L or for L d for all Ad d L or for L d for all or H or H K K 7

26 . Fudameal lemma of Nay-ma ad Pearo Le ad be robably drbuo meaure o ree o σ-fe meaure ad oder he e Eee E have dee ad wh H: v. K For here e a e ad oa k uh ha Ad a k b k Suffe odo for a mo owerful e If afe he odo a ad b for ome k he he mo owerful e for eg aga a level. : Neeary odo for a MP If mo owerful e a level he for ome k afe b ad afe a ule here e a e wh ower ad of ze le ha. Proof oder ad aume ha ad. a be rewre a 8 a umulave drbuo fuo ad o-reag ad ouou o he rgh ad lm lm. I afe ha

27 9 Le be uh ha ad oder he e defed by he ze of If he ad doe o deed o value of. Of hoe : k uque f here doe o e a erval uh ha a oa o.e. ad } : { he d whh ha robably. Aume ha * ay oher e a level a mo wh *. Le be a uably hoe ae dvded o: } : { * } : { * If * be mulled by k he he reul wll be o-egave

28 k k Now * k d * k d d ad d d d * * Le * be mo owerful a level for eg H: v. K: ad le afy a ad b h ae odered a he mo owerful amog all e a level. akg o K { : } ad le S K ; o roof ha S * uoe ha k o S * { } o K o S * k d S h a orado; hee ha more owerful ha * S o e K he owerful e gve uquely a... * a be defed arbrarly o K * f ad aordg o he roof o oble ha defe o be a oa. If here e o e of ower a * level le ha he he mo owerful e afe. orollary Le deoe he ower of he mo owerful e he ule. H: v. K : a level where Proof

29 Se he level e gve by ha ower le be he mo owerful e o ha f e he mo owerful e ad mu afe k o he e k K { : } : K K : k ad oe e uh ha k.. Drbuo wh moooe lkelhood rao Nayma &Pearo geerally o eaded o volve he UMP e he ae ha. A eeo he ae ad he yem of drbuo wh moooe lkelhood rao; h ae Ney-ma &Pearo heorem ued o oru he e H : v. K: or : H v. H :. Defo { he real-arameer famly of drbuo } for whh here a domag σ- fe meaure : dey alled yem wh moooe d d lkelhood rao f here e a real-valued fuo uh ha for ay he rao a o-reag fuo of. heorem Le he radom varable ha robably dey wh moooe lkelhood rao for eg H: v. K : here e a UMP e ad gve by

30 where ad are deermed bye Proof he ower fuo rly reag for all o for whh. For all he e gve by a ad b UMP for eg H : v. K : a v For ay level. he e gve by ad d mmze he robably of a error of he fr kd amog all e afyg d. le ad be wo hoe o where gve ad Aordg o Ney-ma &Pearo heorem here e UMP e k ad k afed Now f k f k Le be he mo owerful for eg H : v. K : where level k f Aordg o orollary : ad k f d a

31 f ad a e a level h lead o he e whh mamze he ower amog all e afyg ad deede of where wa arbrarly hoe he ad he e he ame he UMP e. e he mo owerful for eg H : v. K : where a level If he e H : v. K : wa rerbed a level he lead o he e wh he ame ad deede of. Defo uformly mo owerful e * * afe *..5 Oe-arameer eoeal famly A gle-arameer eoeal famly e of robably drbuo whoe robably dey fuo ha he form e A h. Aordg o Faorzao rera a uffe a of he drbuo hu for eoeal famly here e a uffe a whoe dmeo equal he umber of arameer o be emaed. If A moooe fuo he lead o yem wh moooe lkelhood rao where A rly reag ad e{ A A } orollary reag. I oe-arameer eoeal famly wh Amoooe arameer here e UMP e

32 H: v. H: Or H: v. H: o ad gve by ad afe E.6 Veor arameer eoeal famly he gle-arameer defo a be eeded o a veorarameer... l. A famly P of drbuo ad o be from -dmeoal eoeal famly f he drbuo have dee of he form uh ha wh ree o a σ-fe meaure μ P defed by e h. Here he ad are he real-valued fuo of he arameer are he real-valued a; a o he amle ae. he veor-arameer eoeal famly ad o be aoal form f for all. heorem If he eoeal famly wh aoal form he a... are lear deede he a redue he ze of he aural arameer f he dmeo are o redued he he arameer... udefed. heorem 5 Le be drbued aordg o eoeal famly wh full rak he form e{ U r j j } h j he

33 A radom varable... ha drbuo wh dey j j r e{ } wh ree o σ-fe meaure. j U he odoal drbuo of... gve ha dey u r U e{ } wh ree o meaure. A geeralzao of he Ney-ma Pearo Lemma Le f f... f f m m be real-valued meaurable fuo whh are egraed wh ree o σ-fe ad uoe ha for gve oa : afyg f d d f... m A... Le be a la of ral fuo for whh A hold he U m here e a ral fuo Amog all member of here e uh ha u f f d m d m A A uffe a for a member of o mamze f m d he eee of oa... m uh ha k f m k m m B f m k If a member of afe B whk all ral fuo afyg f f f f d... m f he mamze f m d ad amog v he e M { d... d a ral fuo } ove ad loed m ad f... a er o of M he here e oa... m k k ad m 5

34 a ral fuo afyg A ad B ad a eeary odo for o mamze f m d ha afe B a.e. μ orollary 6 Le... m m be robably dee wh ree o σ-fe meaure ad le. he here e a e uh ha d d ule m k Proof Ug he duo over m: m For m= he reul redue o orollary m almo everywhere μ. Aume ha he aeme afe ad for ay e of m drbuo ha j {... m} uh ha d j m ad d j j j j... m ad Wh h aumo he o... m a er o of M where M { d... m a ral fuo } Wh he geeral Ney-ma Pearo heorem oble o oru UMP e H: or v. K : heorem 7 Le ha drbuo he oe-arameer eoeal famly wh dey e{ A } h A reag he H: v. K: For eg he hyohe or level here e a UMP e gve by whe whe A whe or. a 6

35 Where ad are deermed by B he e gve by A mmze afyg B for all ad amog all e For he ower fuo of h e ha a mamum a o ad dereae rly a ed away from eher dreo. A eeo he uao Lemma 8 Le. ad afy he eoeal yem he revou heorem. Le rly reag for all. If ad * are wo e afyg A * * * * rgh of ; equaly If * ad * o he ha ad a lea oe rog * * * ad afy A ad B from he revou heorem ad almo everywhere. Uformly mo owerful e doe o e for eg he hyohee K: or ad H: or v. K : alway oble o oru UMP e for H: v. K: * H: v. he reao uh ha he form whh mamze ad mmze he ower fuo for reevely. herefore e UMP ubaed e of hee hyohee. ad 7

36 .7 Ubaede for hyohe eg Suoe ha we have he hyohe H: v. H H: whh he ower fuo afe f f H K K ubaed e uh ha for he dea o rer he la of ubaed e ad earh for he roge amog hem. Uformly mo owerful e ubaed wheever e e ower a o fall below ha of he e. If he ower fuo a ouou fuo of θ he ubaed e mu ake oly he value α o he ommo boudary of H ad.e. K H o θ ha are o or lm o of boh he boudary of H ad K. Lemma H ad K K he e of uh a e ad o be Smlar o If he drbuo are uh ha he ower fuo of every e ouou ad f UMP amog all e whh are mlar o he boudary of amog all ubaed e a level α. H ad K he alo UMP Proof he la of e afyg oa he la of ubaed e e UMP amog all ubaed e. O he oher had e boudary ad ha afe K ubaed. mlar o he Eoeal famle wh aural arameer have ouou ad dffereable ower fuo 8

37 heorem Le ha drbuo wh dey e{ } h he e he form or I UMP ubaed e for eg H: v. K: or are deermed by. a level α where Proof Se he ower fuo of ay e he drbuo of eoeal ye ouou ad dffereable he revou lemma a be ued he ommo boudary of { H ad K he e } Smlary o he boudare Lookg for he e whh mamze he ower fuo for ome fed o } From heorem 7 he e he form { whe whe A where B whe or. Mmze he ower fuo for ad Ve-vere mamze h ower fuo amog all e afyg B. heorem Le ha drbuo wh dey e{ } h he e he form or 9

38 I he UMP ubaed e for eg H: v. K: a level α where he oa are deermed by ad e wh uefed uae arameer oder he eoeal famly he form k e{ U } wh ree o... Ad he hyohee : : H v K H : or v K : H : v K : or : : H v K We hall aume ha he arameer ae Θ ove ad o oaed a lear ae of dmeo le ha k ad ha he o are o o he boudary o ha : = : ; From heorem 5 kow ha he drbuo of he uffe a U U... ha dey k k k U u e{ u } Ad odoal dey of U / wh ree o U u e{ u} wh ree o So ha oble o oru he UMP e for eg H he form

39 u whe whe whe u u u Ad for eg H here e UMP e he form u u or u u u We eed o roof ha hee odoal e are UMP amog all ubaed e f ha drbuo from he la { } e mlar o he boudary wh ree o wh ree o Θ f Defo e mlar wh ree o { } f Le be a uffe a for he drbuo of gve doe o deed o ay arbrary e afe alo mlar e wh ree o e afyg almo everywhere ad o be e wh Nay-ma ruure he la of drbuo alled bouded omlee f for all lmed meaurable fuo f he form f f hold almo everywhere. Lemma Le be a radom varable wh drbuo from he famly { } ad le be a uffe a for he all mlar e wh ree o have Ney-ma ruure wh ree o oly f he la of drbuo { } of uffe a bouded omlee Proof

40 Le be bouded omlee famly ad aume ha mlar e Le be lmed fuo afyg From he bouded omleee of almo everywhere ha Nayma ruure. heorem Le be a radom varable drbued aordg o } e{ k j j j wh ree o μ he he uffe a... k ha drbuo of ye } e{ k j j j wh ree o υ ad he famly } { bouded omlee f oa o-degeerae k-dmeoal reagle. Proof heorem 6 e u u u u UMP ubaed e for eg he hyohe : H v. : K a level α where ad are deermed by U ad ha e u u u or u u u UMP ubaed e for eg he hyohe : H v. : K a level α where ad are deermed by U U U ad U.

41 heorem 7 Le ha drbuo of ye } e{ j k j j U wh ree o μ. aume ha h U V doe o deed o f he UMP ubaed e for eg hyohe : H v. : K a level α where ad are deermed by V ad f u h reag υ for every. f u h lear where b u a u h he e or UMP ubaed e for eg he hyohe : H v. : K a level α where ad are deermed by V ad V V V Proof u h reag for every V V V V V V ad are deede of ad are oa. b u a u h U U U a b V V a b V a b V V a b V

42 V V b V b V V V for ad V deede of..8 Lea favorable drbuo Aume ha we have he hyohe of eg H v. K : : H: v. K: ad oder he roblem ; he dea o relag H whh uoed o rovde o formao oerg by a equvale mle hyohe H : where a robably drbuo over ad a drbuo wh dey d orug h roblem requre: a σ-algebra τ be defed over he arameer ae or a ad a robably drbuo Λ be defed over τ where Λ mu be formave or lea rovde a a lle hel for he uroe of relag hyohe H by H. For all Λ drbuo over τ le deoe he mamum ower for eg H v. K a level. Defo he drbuo Λ ad o be lea favorable a level f for ay oher drbuo he equaly heorem o hold. Le τ be a gve σ-algebra over uh ha he dee are joly meaurable θ ad. Suoe ha over here e a robably drbuo Λ uh ha he mo owerful level-α e for eg H v. K orgal hyohe H. he e mo owerful for eg H v. K. of ze le ha α alo wh ree o he

43 If he uque mo owerful level-α e for eg H v. K alo he uque mo owerful e of H v. K a level α. he drbuo Λ he lea favorable. orollary Suoe ha Λ a robably drbuo over ad ha be a e uh ha wh. Le q k q k d d he a mo owerful level-α e for eg H: v. q K: f u 5

44 haer he urvaure eoeal famle We kow ha f F a eoeal famly he he mehod meoed he revou haer a be ued whou ay roblem ad ow he queo wha f hee famle are oeoeal? here a quay ha meaure how early eoeal a arbrary oe-arameer famly. h quay alled a aal urvaure ad wa fr rodued by Bradley Efro. he aal urvaure deally zero f F eoeal famly ad greaer ha zero for a lea ome value of he arameer for o-eoeal famly. I wa oed ha he famly wh a mall urvaure ejoy he good roere of eoeal famle bu bg urvaure overhrow hee roere for eamle he MLE o loger a uffe a for oeoeal famle. he amou of formao we lo a be ereed a erm of urvaure.. he geeral defo of aal urvaure Le F= {f θ θ Θ} be a arbrary famly of dey fuo deed by he gle arameer θ Θ. Le lθ deoe he logarhm of f θ ad he fr ad eod aral dervao wh ree o θ by uoe ha he dervave e ououly ad a be uformly domaed by egrable fuo a eghborhood of he gve θ. he mome relaoh l l l Le M θ be he ovarae mar of l l l l l l l l So ha he aal urvaure of F a θ defed a / / = / aumg ha < θ< ad ν θ< 6

45 I advaageou o meo ha he aal urvaure of drbuo relaed drely wh he geomeral urvaure. A we wll ee for urved eoeal famle he aal urvaure ealy equal he ordary geomer urvaure of he le.. he geomeral urvaure Le L be a urved le Euldea k-ae Ε k defed a L={ η θ θ Θ} where Θ a erval of he real le ad η θ he veor Ε k for eah θ. he urvaure of L a θ wh ree o he er rodu / / defed a M θ mar defed a Subug k= θ= η θ = Y he he urvaure of L redue o Y Y / If he rae of hage of dreo of L wh ree o ar-legh alog he urve dereag he he radu of he rle age o L a Y reag. I a be deoed a =/γ where ad o be he radu of urvaure. Now we omue he aal urvaure for urved eoeal famle: Suoe we have a eoeal famly wh dey formulaed a e D h η H ad H he arameer ae a wa meoed he fr haer Le L= {η θ θ Θ} be a oe-arameer ube he eror of H where η θ ououly we dffereable fuo of θ Θa erval of he real le. Defe he famle F o be urved ube of a large k-arameer eoeal famly. Le be defed a f e D h he {f θ θ Θ} urved eoeal famly 7

46 8. he roere of he urvaure he fr mora roery of he aal urvaure ha doe o deed o he arameerzao o ha f we ue ead of arameer θ a rly moooe we dffereable fuo of θ he he urvaure wll o hage for he ew arameerzao ad wll ay he ame boh ae. he eod roery he varae of he aal urvaure uder ay mag o uffe a. If we aumed ha we have a uffe a for he arameer θ he he lkelhood fuo ug he uffe a wll ay he ame a he lkelhood ug he whole daa whh mea ha he aal urvaure wll o hage eher. he urvaure eamle. urved ormal famly haer We had a wo-parameer eoeal famly he form e f ~ N l l l f l f l f l l l l Se a l l

47 9 l l 6 8 Se l v 9 l Now he aal urvaure 7 7 / 6 6 For he roere Aume ha we have a ew arameerzao for o ha ~ / rly moooe we dffereable fuo of ~ ~ ~ N ~ ~ ~ ~ ~ ~ ~ 7 ~ 7 / ~ 6 6 he ame value wh he ew arameerzao ad deede of he arameer boh ae.

48 Referee Bradley Efro. 975: Defg he urvaure of a Saal Problem Wh Alao o Seod Order Effey. he Aal of Sa Vol. No Saford Uvery alfora. Lehma E. L. 8: eg Saal Hyohe. Srger New York. Lehma E. L. : heory of Po Emao. Srger New York. Lee F. Meke K. J. 9: Saal Deo heory. Emao eg ad Seleo. Srger New York. 5