TIGHT BOUNDS ON EXPECTED ORDER STATISTICS

Transcription

1 Probability i the Egieerig ad Iformatioal Scieces, 20, 2006, Prited i the U+S+A+ TIGHT BOUNDS ON EXPECTED ORDER STATISTICS DIMITRIS BERTSIMAS Sloa School of Maagemet ad Operatios Research Ceter Massachusetts Istitute of Techology Cambridge, MA dbertsim@mit.edu KARTHIK NATARAJAN Departmet of Mathematics Natioal Uiversity of Sigapore Sigapore matkb@us.edu.sg CHUNG-PIAW TEO Departmet of Decisio Scieces NUS Busiess School Sigapore bizteocp@us.edu.sg I this article, we study the problem of fidig tight bouds o the expected value of the kth-order statistic k: # uder first ad secod momet iformatio o real-valued radom variables+ Give meas i # µ i ad variaces Var@X i # s i 2, we show that the tight upper boud o the expected value of the highest-order statistic : # ca be computed with a bisectio search algorithm+ A extremal discrete distributio is idetified that attais the boud, ad two closed-form bouds are proposed+ Uder additioal covariace iformatio Cov@X i, X j # Q ij, we show that the tight upper boud o the expected value of the highest-order statistic ca be computed with semidefiite optimizatio+ We geeralize these results to fid bouds o the expected value of the kth-order statistic uder mea ad variace iformatio+ For k, this boud is show to be tight uder idetical meas ad variaces+ All of our results are distributiofree with o explicit assumptio of idepedece made+ Particularly, usig optimizatio methods, we develop tractable approaches to compute bouds o the expected value of order statistics Cambridge Uiversity Press $

2 668 D. Bertsimas, K. Nataraja, ad C.-P. Teo. INTRODUCTION Let X ~X,+++,X! deote 2 joitly distributed real-valued radom variables+ The order statistics of this set is a reorderig of the X i i terms of odecreasig values, expressed as X : {{{ X k: {{{ X : + The lowest- ad highest-order statistics are deoted by X : ad X :, respectively+ Oe of the cetral problems i statistics is to fid, boud, or approximate the expected value of order statistics uder varyig assumptios o the distributio of the radom variables+ For detailed reviews o this subject, the reader is referred I this article, we focus o fidig bouds o the expected value of order statistics uder momet iformatio o the radom variables+ Let X ; u m deote the set of feasible distributios u that satisfies the give momets m for the radom variables+ Defiitio : Z k: is a tight upper boud o the expected value of the kth-order statistic if Z k: sup E k: #; X; u m that is, there exists a feasible distributio or a limit of a sequece of feasible distributios that achieves the upper boud. No other assumptios o idepedece or the type of distributio are made+ I this article, we develop methods to compute Z k: uder first ad secod momet iformatio o the radom variables+ Next, we review some of the classical bouds for order statistics+.. Some Kow Bouds Give idetical meas ad variaces ~ µ, s 2! for the radom variables, oe of the earliest kow bouds for the expected highest-order statistic was derived by ad Hartley ad Uder the assumptio of idepedece, they obtaied the upper boud µ s~! M 2 + exteded this result to the special case of symmetrically distributed radom variables+ For more geeral distributios ~ot ecessarily idepedet or idetically distributed!, Arold ad obtaied a upper boud o the expected value of the kth-order statistic: 2 ( µ i E k: # k ( ( µ i s 2 i µ i + () ~ k! Uder idetical meas ad variaces, this boud reduces to E k: # µ s k k + (2)

3 For this particular case, Arold ad showed that ~2! is tight by explicitly costructig a distributio that achieved the boud+ However, for geeral mea variace iformatio, ~! is ot ecessarily tight+ proposed a alterative upper boud o the expected value of the highest-order statistic: E : # max µ i i ( s i 2 + (3) This boud is also ot tight uder geeral mea variace iformatio+ I this article, we develop a algorithmic approach to fid ~possibly! tight bouds o the expected value of the order statistic Z k: + We characterize cases for which the boud ca be computed tractably, or we propose simple closed-form bouds that seem promisig+.2. Cotributios TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 669 Our mai cotributios i this article are as follows: + I Sectio 2 we fid the tight upper boud o the expected value of the highest-order statistic Z : uder mea variace iformatio o the radom variables+ A efficietly solvable bisectio search approach is developed to compute Z : + A discrete extremal distributio is idetified that attais the tight boud+ Two simple closed-form bouds for the expected highest-order statistic are proposed+ Uder additioal covariace iformatio, we propose a semidefiite programmig approach to fid the tight boud o the expected highest-order statistic+ 2+ I Sectio 3 we exted the bisectio search method to obtai bouds o the expected value of the geeral kth-order statistic uder mea variace iformatio+ For k, we show that the boud is tight uder idetical meas ad variaces+ For geeral mea variace iformatio, the boud foud with the bisectio search method, although ot ecessarily tight, is at least as strog as ~!+ 3+ I Sectio 4 we provide computatioal experimets to test the performace of the differet bouds+ 2. BOUNDS ON EXPECTED HIGHEST-ORDER STATISTIC We first compute the tight upper boud o the expected highest-order statistic Z : uder mea variace iformatio o the radom variables+ The mea ad variace iformatio o the radom variables are deoted as m ~ µ,+++,µ! ad s 2 ~s 2,+++,s 2!+ The set of feasible distributios satisfyig these momet restrictios is represeted by X ; u ~ m, s 2!+ For simplicity of presetatio, we will assume that all of the s i are strictly positive+ As discussed later, this coditio ca i fact be relaxed+

4 670 D. Bertsimas, K. Nataraja, ad C.-P. Teo The approach to compute the tight upper boud o the expected value of the highest-order statistic is based o a covex reformulatio techique, iitially proposed by Meilijso ad ad developed later i Bertsimas, Nataraja, ad The reformulatio is based o the observatio that the highest-order statistic X : is a covex fuctio i the X i variables+ We review the key ideas of this reformulatio ext+ Theorem ~Bertsimas et The tight upper boud o the expected value of the highest-order statistic Z : give X ; u ~ m, s 2! is obtaied by solvig Z : mi where x max~0, x!. z z : ( sup E i z i #, (4) X i ; ui ~ µ i, s 2 i! Sketch of Proof: We first show that ~4! provides a upper boud o Z : + To see this, ote that we have the followig iequality for each variable X i : X i z i ~X i z i! z : i z i # + Sice the right-had side of this iequality is idepedet of the particular i, we have X : z : i z i # + Takig expectatios ad miimizig over the z i variables, we obtai the best upper boud: E : # mi z : ( E i z i # + z Optimizig over distributios with give mea variace iformatio, we obtai a upper boud: Z : mi z : ( z sup X i ; ui ~ µ i, s 2 i! E i z i # + Note that the ier problem is optimizatio over probability distributios of sigle radom variables u i, sice o cross-momet iformatio is specified+ For a proof that the boud is tight, the reader is referred Alteratively, we costruct a extremal distributio i Theorem 3 that attais the boud+

5 TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 67 The solutio for the ier problem i ~4! is i fact kow i closed form We ow outlie a simple proof for this boud+ Propositio : The tight upper boud o the expected value E i z i # give X i ; ui ~ µ i, s i 2! is sup E i z i # X i ; ui ~ µ i, s 2 i! i z i M~ µ i z i! 2 s 2 i #+ (5) Proof: We have the basic i z i # 2 ~X i z i 6X i z i 6!+ Takig expectatios, we obtai E i z i # 2 ~E u i z i # E ui 6X i z i 6!, X i ; ui ~ µ i, s i 2! 2 ~ µ i z i M~ µ i z i! 2 s i 2!, ~from the Cauchy Schwarz iequality!+ Furthermore, this boud ca be show to be tight sice it is attaied by the distributio 2 i M~ µ i z i X i z!2 s 2 i, w+p+ p µ i z i 2 M~ µ i z i! 2 s i z i M~ µ i z i! 2 s i 2, w+p+ p 2 µ i z i M~ µ i z i! 2 s i 2 Usig this closed-form boud, we ow show that the tight upper boud o the expected highest-order statistic ca be foud by solvig a uivariate covex miimizatio problem+ Theorem 2: The tight upper boud o the expected value of the highest-order statistic Z : give X ; u ~ m, s 2! is obtaied by solvig the strictly covex uivariate miimizatio problem Z : mi f : ~z! mi z ( (6) z z i z M~ µ i z! 2 s i 2 # +

6 672 D. Bertsimas, K. Nataraja, ad C.-P. Teo Proof: Combiig Theorem ad Propositio, the tight upper boud o the expected highest-order statistic is Z : mi z : ( z i z i M~ µ i z i! 2 s 2 i # + (7) We ext show that ~7! ca be simplified to a sigle-variable optimizatio problem+ Let z be a optimal solutio to ~7! ad let z : deote the highest-order statistic+ Note that the secod term, ( _ 2@µ i z i M~ µ i z i! 2 s 2 i #, is decreasig i z i + Hece, for ay i with z i: z :, by icreasig z i: up to z : the first term remais uaffected while the secod term decreases, thus reducig the objective+ Sice we are miimizig the objective, the optimal solutio will set all of the z i values equal to z : + It ca be easily checked that f : is a strictly covex fuctio, implyig that the fuctio has a uique global miimum+ The optimal decisio variable z i ~6! hece satisfies the first-order coditio obtaied by settig the derivative ]f : ~z!0]z to zero: ]f : ~z! ( ]z z µ i M~ µ i z! 2 s i 2 ~ 2! 0+ (8) Remark: Our result ca be viewed as a extesio of the boud from Lai ad ad I their case, uder completely kow margial distributios X i ; u u i, they obtai the followig tight boud o the highest-order statistic: sup X i ; u u i, i E : # mi z ( z Note that this result follows also from Meilijso ad E i z# + (9) 2.. A Extremal Probability Distributio We costruct a -atom discrete distributio that satisfies that mea variace requiremets ad attais the boud i ~6!+ Theorem 3: Let z deote the optimal miimizer to (6). A -atom extremal distributio for X that achieves the upper boud ad satisfies the mea variace requiremets is X X ~ j!, w.p. p j 2 µ j z M~ µ j z! 2 s j 2 for j,+++,,

7 TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 673 where X ~ j! ~ j ~X! ~ j,+++,x!! is expressed as ~ j X z M~ µ! i z! 2 2 s i if i j i z M~ µ i z! 2 2 s i if i j+ Proof: From the defiitio, it is clear that the p j values deote a probability measure sice p j 0 for all j ad ( j p j ( j 2 µ j z M~ µ j z! 2 s i ~from ~8!! It ca be verified for this -atom distributio that ~ j E i # ( p j X! i µ i, j i,+++,, ~ j Var i # ( p j ~X! i µ i! 2 s 2 i, j i,+++,+ Furthermore, the maximum amog the radom variables for the jth atom is attaied by X j ~ j! + Thus, E : # ( j ~ j p j X! j z ( j j z M~ µ j z! 2 s 2 j # f : ~z!+ This -atom distributio attais the upper boud o the expected value of the highestorder statistic ad satisfies the mea ad variace requiremets+ This verifies that the boud i Theorem is tight Solutio Techiques I geeral, it does ot seem possible to fid Z : i closed form+ A special case uder which this is possible is discussed ext Idetical Mea ad Variace For idetical mea variace pairs ~ µ, s 2!, solvig ~8! yields the optimal value for z : z µ s 2 2M +

8 674 D. Bertsimas, K. Nataraja, ad C.-P. Teo Substitutig this ito ~6! yields the tight boud sup X i ; u ~ µ, s 2! i E : # µ sm + (0) Note that this is exactly ~2! obtaied by Arold ad for k + A distributio that attais this boud is radomly selectig elemets without replacemet from the set i which oe elemet has the value µ sm ad the remaiig elemets have the value µ s M Geeral Mea Variace Pairs For the geeral case, we outlie a simple bisectio search algorithm to fid Z : + Descriptio of the Algorithm + Iitialize z l ad z u such that ]f : ~z l!0]z 0 ad ]f : ~z u!0]z 0 ad e 0 to a give tolerace level+ 2+ Let z ~z l z u! While 6]f : ~z!0]z6 e, do: ~a! If ]f : ~z!0]z 0, set z u z; or else set z l z+ ~b! Go back to Step Output Z : f : ~z!+ We propose two simple upper ad lower bouds z u ad z l o the rage of the optimal z to iitialize the algorithm+ Cosider the problem of fidig a z u such that f ' ~z u! 0+ Oe such z u is costructed such that each term o the left-had side of ~8! cotributes at least a fractio ~ 2!0: which reduces to z u µ i M~ µ i z u! 2 s 2, i,+++,, 2 i z u µ i s i 2 2M, i,+++,+ We choose z u as z u max i µ i s i 2 2M + () Similarly, a lower boud z l ca be foud such that z l µ i M~ µ i z l! 2 s 2, i,+++,+ 2 i

9 A z l that satisfies this coditio is z l mi i µ i s i 2 2M + (2) Our computatioal tests idicate that these values of z u ad z l lead quickly to the tight boud New Closed-Form Bouds Based o the two ed poits, we ow propose simple closed-form bouds o the expected value of the highest-order statistic+ Theorem 4: Two closed-form upper bouds o the expected value of the highestorder statistic give X ; u ~ m, s 2! are 2 ( µ i µ i max i µ i 2 2 i 2M s s i 2 ~2! max i µ i 2 2M s i, (3) 2 ( µ i µ i mi i µ i 2 2 i 2M s s i 2 ~2! mi i µ i 2 2M s i + (4) Proof: Substitute z z l ad z z u i ~6! respectively+ Note that ~3! ad ~4! reduce to the tight upper boud ~0! o the expected highest-order statistic for radom variables with idetical mea variace pairs Extesios TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 675 We ow exted the results to the case where some of the s i 2 0 ~i+e+, X i is determiistic!+ Without loss of geerality, we assume that exactly oe variable is determiistic sice the case with multiple costats ca be reduced to this case by choosig the maximum of the costats+ Give radom variables with strictly positive variaces ad a costat K, we wat to fid the tight upper boud o E :, K!# + By itroducig a extra decisio variable, z, for the term K, ~4! reduces to sup E :, K!# X; u ~ m, s 2! mi z : ( z i z i M~ µ i z i! 2 s 2 i # ~K z! +

10 676 D. Bertsimas, K. Nataraja, ad C.-P. Teo Usig a argumet similar to Theorem 2, it ca be checked that the optimal solutio will set all of the z i values the same at a value greater tha or equal to K+ Hece, the tight upper boud o the expected highest-order statistic is sup E :, K!# mi X; u ~ m, s 2! z K z ( which reduces to the costraied versio of ~6!: sup X; u ~ m, s 2! i z M~ µ i z! 2 s i 2 #, (5) E :, K!# mi f : ~z!+ (6) z K The tight upper boud ca be foud by a modified bisectio search method: + Solve the ucostraied versio of ~6! with bisectio search to fid z + 2+ Output f : ~max~z, K!!+ We propose usig the followig two closed-form bouds i this case: ad f : max max i µ i 2 2M s i, K (7) f : max mi i µ i 2 2M s i, K + (8) 2.4. Extesios to Additioal Covariace Iformatio I this subsectio, we propose a algorithmic approach to fid the tight upper boud o the expected value of the highest-order statistic uder covariace iformatio+ Give the mea ad covariace matrix for the radom variables X ; u ~ m,q!, the tight upper boud is computed by fidig a distributio u that solves Z : sup u E : # s+t+ E # m E ' # Q mm ', E # + (9) Here I ~X! ifx ad zero otherwise represets the idicator fuctio+ This problem has bee well studied uder the class of momet problems i ad Karli ad To solve ~9!, we costruct the dual problem by itroducig variables y, Y, ad y 0 for each of the momet costraits+ The dual problem is formulated as

11 TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 677 Z mi~ y ' m Y{~Q mm '! y 0! s+t+ y ' X X ' YX y 0 X :, X + (20) The costraits i ~20! imply the oegativity of a quadratic fuctio over + By takig the expectatio of the dual costraits, it is easy to see that Z Z : + Furthermore, showed that if the covariace matrix Q 0 is strictly positive defiite, the Z Z : + Uder this assumptio, the covexity of X : implies that the tight upper boud o the expected highest-order statistic is Z : mi~ y ' m Y{~Q mm '! y 0! s+t+ y ' X X ' YX y 0 X i, i,+++,, X + (2) Let e ~i! deote a uit vector with the ith compoet e i ~i! ad zero otherwise+ The equivalece betwee the global oegativity of a quadratic polyomial ad the semidefiite implies that ~2! ca be rewritte as Z : mi~ y ' m Y{~Q mm '! y 0! Y ~ y e s+t+ i!02 ~ y e i! ' 02 y 0 0, i,+++,+ (22) Here A 0 deotes the costrait that the matrix A is positive semidefiite+ Formulatio ~22! is a semidefiite optimizatio problem that ca be solved withi e 0 of the optimal solutio i polyomial time i the problem data ad I practice, stadard semidefiite optimizatio codes such as ca be used to fid the tight upper boud o the expected highest-order statistic uder covariace iformatio+ 3. BOUNDS ON EXPECTED k TH-ORDER STATISTIC I this sectio, we geeralize our results to fid bouds o the expected value of the kth-order statistic for k uder mea variace iformatio o the radom variables; that is, Z k: sup E k: #+ X; u ~ m, s 2! Our results are based o the simple observatio that ( X i: i k X k: k + (23) We fid tight bouds o the expected value of the right-had side of ~23! to obtai bouds o the expected value of the kth-order statistic+

12 678 D. Bertsimas, K. Nataraja, ad C.-P. Teo Theorem 5: The tight upper boud o the expected value of the sum of the kth- to th-order statistic give X ; u ~ m, s 2! is obtaied by solvig sup X; u ~ m, s 2! E u ( i k X i: mi ~ k!z ( z i z M~ µ i z! 2 s 2 i # + (24) Proof: Usig the result from Bertsimas et the upper boud o the sum of the expected value of the kth- to th-order statistic is sup X; u ~ m, s 2! E u ( i k X i: mi z ( i k z i: ( i z i M~ µ i z i! 2 s 2 i # + As earlier, ~25! ca be reduced to a sigle-variable optimizatio problem+ To see this, let z be a optimal solutio to ~25!+ For ay l k with z l: z k:, we ca icrease z l: to z k: sice the first term is uaffected ~( i k z i: is uaffected by chage i z l:, for l k, provided z l: z k:! while the secod term decreases i z l: + Hece, we have z l: z k: for l k+ Furthermore, for l k with z l: z k:, by decreasig z l: to z k: the first term decreases at a rate of while the secod term icreases at a rate of at most + Sice we wat to miimize our objective, we have z l: z k: for l,+++,+ Usig ~23! ad Theorem 5, we ow obtai a boud o the expected kth-order statistic+ Theorem 6: A upper boud o the expected value of the kth-order statistic Z k: give X ; u ~ m, s 2! is obtaied by solvig Z k: mi z f k: ~z! mi z z i z M~ µ i z! 2 s 2 i # + 2~ k! Note that the ocovex structure of the kth-order statistic for k implies that ~26! is ot ecessarily tight for geeral mea variace pairs+ However, ~26! is at least as tight as ~! proposed by Arold ad This follows from observig that they obtaied their boud by boudig ~23!, although ot i the tightest maer+ A special case uder which ~26! is tight is described ext+ 3.. Idetical Mea ad Variace For idetical mea variace pairs ~ µ, s 2!, ~26! yields the optimal value for z : z µ s 2k 2 2M~k!~ k! + (25) (26)

13 Substitutig this ito ~26! yields sup X i ; u ~ µ, s 2!, i E k: # µ s k k + (27) This is exactly ~2! obtaied by Arold ad To see that ~27! is tight, cosider a distributio obtaied by radomly selectig elemets without replacemet from the set i which k elemets have the value µ sm ~k!0~ k! ad the remaiig k elemets have the value µ sm ~ k!0~k!+ It is easy to verify that this distributio attais the boud as described earlier Geeral Mea-Variace Pairs For the geeral case, we propose the use of the bisectio search algorithm to fid the boud o the expected kth-order statistic by solvig mi z f k: ~z!+ The lower ad upper bouds o the rage of the optimal z to iitialize the bisectio search method i this case reduces to 2k 2 z u max i µ i s i (28) 2M~k!~ k! ad TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 679 2k 2 z l mi i µ i s i (29) 2M~k!~ k! + Theorem 7: Two closed form upper bouds o the expected value of the kth-order statistic give X ; u ~ m, s 2! are Z k: f k: 2k 2 max i µ i s i (30) 2M~k!~ k!, Z k: f k: 2k 2 mi i µ i s i (3) 2M~k!~ k! + 4. COMPUTATIONAL RESULTS I this sectio, we evaluate the quality of the various bouds proposed i this article+ The first example is a applicatio of the highest-order statistic boud i a fiacial cotext+ The secod example is a simulatio experimet to compare the performace of the bouds for the geeral kth-order statistic+ The computatios were coducted o a Petium II ~550 MHz! Widows 2000 platform with the total computatioal time uder a miute+ 4.. Applicatio i Optio Pricig Oe of the cetral questios i fiacial ecoomics is to fid the price of a derivative security give iformatio o the uderlyig assets+ Uder a geometric Brow-

14 680 D. Bertsimas, K. Nataraja, ad C.-P. Teo ia motio assumptio o the prices of the uderlyig assets ad usig the o-arbitrage assumptio, the Black formula provides a isightful aswer to this questio+ Assumig o arbitrage, but without makig specific distributioal assumptios, Bertsimas ad ad Boyle ad derived momet bouds o the prices of optios+ Our particular focus is o fidig bouds o the price of a optio kow as the lookback optio uder momet iformatio o the asset prices+ Let x, x 2,+++,x deote the price of a asset at differet times+ A simple lookback Europea call optio o these assets with strike price K 0 has a payoff of max~x : K,0!+ Let r deote the risk-free iterest rate ad T deote the maturity date+ Uder the o-arbitrage assumptio, the price of the lookback optio is P~K! e rt E : K,0!#, (32) where the expectatio is take over the martigale measure+ Clearly, the price of this optio depeds o the highest-order statistic+ Uder mea ad variace iformatio o X i, Boyle ad proposed the followig upper boud o the price of the lookback optio: P~K! e rt ( i K M~ µ i K! 2 s 2 i #+ (33) We use the results from Sectio 2 to fid the best bouds o P~K!+ Note that although the asset prices are oegative i practice, we do ot model this explicitly here to compute our bouds+ The specific lookback optio-pricig example is take from A upper boud o the price of a Europea call lookback optio over 0 time steps is calculated+ The risk-free iterest rate ~r! is 5% ad the time to maturity ~T! is year+ Table provides the mea ad variace iformatio of the asset prices over the 0 periods+ The bouds o the optio price are computed for strike prices K from 70 to 40 i steps of 0+ Table 2 provides six bouds uder mea variace iformatio ad Table. Mea Variace Data o Asset Prices Asset Price X i Mea µ i Variace s i 2 Asset Price X i Mea µ i Variace s i Source:

15 Table 2. Upper Boud o Lookback Call Optio Price Boud Tight mea variace boud ~6! Our closed-form boud ~7! Our closed-form boud ~8! Boyle ad Li s boud ~33! Arold ad Groeeveld s boud ~! Ave s boud ~3! Tight mea variace covariace boud ~22! K Source: 68

16 682 D. Bertsimas, K. Nataraja, ad C.-P. Teo a additioal boud uder covariace iformatio+ For the last boud, we assumed that the asset prices were ucorrelated ad solved ~22! with the semidefiite optimizatio code SeDuMi+ From Table 2, it is observed that Boyle ad Li s boud is very loose for small values of K+ O average, our proposed closed-form boud ~7! outperforms both Arold ad Groeeveld ad Ave boud, respectively+ Although the closed-form boud ~8! is weaker for smaller K, it is i fact tight for larger K, idicatig its usefuless+ I Figure, we provide the graphical compariso of the bouds ~excludig Boyle ad Li boud, which is tight oly for large K! Simulatio Test The secod example is a simulatio test to compare the relative performace of the differet bouds uder radomly geerated momet iformatio+ We cosider 30 radom variables+ The mea variace pairs for each radom variable were idepedetly chose from a uiform distributio with µ i ; 50# ad s i 2 ; 400#+ Oe hudred mea variace pairs were sampled i these rages ad Figure. Upper boud o lookback call optio price+ From

17 TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 683 the bouds o the expected order statistics were computed+ For each closed-form boud, we evaluate the relative percetage error: Closed form boud Bisectio search boud Percetage error 00%+ Bisectio search boud For the highest-order statistic, the percetage error of the bouds are provided i Figure 2 ad Table 3+ Note that i this case, the bisectio search method fids the tight boud Z : + I this case, our closed-form boud ~3! performs the best ad boud ~4! is relatively weaker+ We ext cosider the results for a lower-order statistic+ Sice the upper boud for the lowest-order statistic Z : from ~23! simply reduces to ( µ i 0, we use the secod lowest-order statistic Z 2: to compare the bouds+ For this case, the bisectio search method does ot guaratee fidig the tight boud+ The results obtaied are preseted i Figure 3 ad Table 4+ For this case, our closed-form boud ~4! is Figure 2. Deviatio of closed-form bouds from tight boud o expected highestorder statistic+

18 684 D. Bertsimas, K. Nataraja, ad C.-P. Teo Table 3. Statistics of Deviatio of Closed-Form Bouds for Expected Highest-Order Statistic Boud Mea % Error Std+ Dev+ % Error Our closed-form boud ~3! Our closed-form boud ~4! Arold ad Groeeveld s boud ~! Ave s boud ~3! observed to be tightest amog the closed-form bouds, with a average percetage error of about %+ The simulatio results seem to idicate that the two closed-form bouds perform well i reasoable settigs+ Iterestigly, i each of the two simulatios, the best-closed form bouds were observed to be oe of our bouds+ Although Figure 3. Deviatio of closed-form bouds from bisectio boud o secodorder statistic+

19 TIGHT BOUNDS ON EXPECTED ORDER STATISTICS 685 Table 4. Statistics of Deviatio of Closed-Form Bouds for Expected Secod-Order Statistic Boud Mea % Error Std+ Dev+ % Error Our closed-form boud ~30! Our closed-form boud ~3! Arold ad Groeeveld s boud ~! cases ca be costructed for which both of the bouds are weaker tha either Arold ad Groeeveld s ad Ave s bouds, the results suggest that the bouds are useful+ 5. SUMMARY I this article, we studied the problem of fidig tight bouds o the expected value of order statistics uder first ad secod momet iformatio o the radom variables+ For the highest-order statistic, we showed that the tight upper boud could be foud efficietly uder mea variace iformatio with a bisectio search method ad uder mea variace covariace iformatio with semidefiite programmig+ For the geeral kth-order statistic, we provided efficietly computable bouds ~ot ecessarily tight! uder mea variace iformatio+ Fidig tight bouds for the geeral kth-order statistic uder mea variace ad possibly covariace iformatio is a potetial research area for the future+ Ackowledgmet This research was supported i part by the Sigapore MIT alliace+ Refereces + Adrease, J+ ~998!+ The pricig of discretely sampled Asia ad lookback optios: A chage of umeraire approach+ Joural of Computatioal Fiace 2~!: Arold, B+C+ & Balakrisha, N+ ~989!+ Relatios, bouds ad approximatios for order statistics+ Lecture Notes i Statistics No+ 53+ Berli: Spriger-Verlag+ 3+ Arold, B+C+ & Groeeveld, R+A+ ~979!+ Bouds o expectatios of liear systematic statistics based o depedet samples+ Mathematics of Operatios Research 4~4!: Ave, T+ ~985!+ Upper ~lower! bouds o the mea of the maximum ~miimum! of a umber of radom variables+ Joural of Applied Probability 22: Bertsimas, D+, Nataraja, K+, &Teo, C+-P+ ~2004!+ Probabilistic combiatorial optimizatio: Momets, semidefiite programmig ad asymptotic bouds+ SIAM Joural of Optimizatio 5~!: Bertsimas, D+ & Popescu, I+ ~2002!+ O the relatio betwee optio ad stock prices: A covex optimizatio approach+ Operatios Research 50~2!: Black, F+ & Scholes, M+ ~973!+ The pricig of optios ad corporate liabilities+ Joural of Political Ecoomy 8: Boyle, P+ & Li, X+S+ ~997!+ Bouds o cotiget claims based o several assets+ Joural of Fiacial Ecoomics 46:

20 686 D. Bertsimas, K. Nataraja, ad C.-P. Teo 9+ David, H+A+ & Nagaraja, H+N+ ~2003!+ Order statistics, 3rd ed+ New York: Wiley+ 0+ Gumbel, E+J+ ~954!+ The maximum of the mea largest value ad of the rage+ Aals of Mathematical Statistics 25: Hartley, H+O+ & David, H+A+ ~954!+ Uiversal bouds for mea rage ad extreme observatios+ Aals of Mathematical Statistics 25: Isii, K+ ~963!+ O the sharpess of Chebyshev-type iequalities+ Aals of the Istitute of Statistical Mathematics 4: Jagaatha, R+ ~976!+ Miimax procedure for a class of liear programs uder ucertaity+ Operatios Research 25~!: Karli, S+ & Studde, W+J+ ~966!+ Tchebycheff systems: With applicatios i aalysis ad statistics+ New York: Wiley Itersciece+ 5+ Lai, T+L+ & Robbis, H+ ~976!+ Maximally depedet radom variables+ Proceedigs of the Natioal Academy of the Scieces of the Uited States of America 73~2!: Lo, A+W+ ~987!+ Semi-parametric upper bouds for optio prices ad expected payoffs+ Joural of Fiacial Ecoomics 9: Meilijso, I+ & Nadas, A+ ~979!+ Covex majorizatio with a applicatio to the legth of critical path+ Joural of Applied Probability 6: Moriguti, S+ ~95!+ Extremal properties of extreme value distributios+ Aals of Mathematical Statistics 22: Nesterov, Y+ & Nemirovkii, A+ ~994!+ Iterior poit polyomial algorithms for covex programmig+ Studies i Applied Mathematics 3+ Philadelphia: Society for Idustrial ad Applied Mathematics+ 20+ Parillo, P+A+ ~2000!+ Structured semidefiite programs ad semi-algebraic geometry methods i robustess ad optimizatio+ PhD thesis, Califoria Istitute of Techology+ 2+ Ross, S+M+ ~2003!+ Itroductio to probability models, 8th ed+ New York: Academic Press+ 22+ Scarf, H+ ~958!+ A mi-max solutio of a ivetory problem+ I K+J+ Arrow, S+ Karli, &H+ Scarf ~eds+!+ Studies i the mathematical theory of ivetory ad productio. Staford, CA: Staford Uiversity Press, pp Sturm, J+F+ SeDuMi versio +03+ Available from