# TIGHT BOUNDS ON EXPECTED ORDER STATISTICS

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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, ( _ 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-

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

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+

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