A KERNEL APPROACH TO ESTIMATING THE DENSITY OF A CONDITIONAL EXPECTATION. Samuel G. Steckley Shane G. Henderson

Transcription

1 Proceedings of te 3 Winter Siulation Conference S Cick P J Sáncez D Ferrin and D J Morrice eds A KERNEL APPROACH TO ESTIMATING THE DENSITY OF A CONDITIONAL EXPECTATION Sauel G Steckley Sane G Henderson Scool of Operations Researc and Industrial Engineering Cornell Universtiy Itaca NY 4853 USA ABSTRACT Given uncertainty in te input odel and paraeters of a siulation study te goal of te siulation study often becoes te estiation of a conditional expectation Te conditional expectation is expected perforance conditional on te selected odel and paraeters Te distribution of tis conditional expectation describes precisely and concisely te ipact of input uncertainty on perforance prediction In tis paper we estiate te density of a conditional expectation using ideas fro te field of kernel density estiation We present a result on asyptotically optial rates of convergence and exaine a nuber of nuerical exaples INTRODUCTION Let X be a real-valued rando variable wit E X < Let Z be soe oter rando variable Te conditional expectation E(X Z is a rando variable tat represents one s best guess (in a certain sense as to te value of X given only te value of te rando variable Z In tis paper we assue tat te rando variable E(X Z as a density wit respect to Lebesgue easure and develop a etod for estiating it Our ain assuptions are tat we can generate iid replicates of te rando variable Z and we can generate iid observations fro te conditional distribution P(X Z = z for any z in te range of Z In tis paper we confine our attention to te case were Z is a real-valued rando variable but we are working on establising analogous results wen Z is a ore coplicated rando object Suc a generalization is iportant because our priary otivation for studying tis proble stes fro te issue of input odel uncertainty Tis for of uncertainty arises wen one is not copletely certain wat input distributions and associated paraeters sould be used in a siulation odel Tere are any etods for dealing wit suc uncertainty; see Henderson (3 for a review Many of tese etods ipose a probability distribution on te input distributions and paraeters For exaple tis is te case in te papers Ceng (994 Ceng and Holland (997 Ceng and Holland (998 Ceng and Holland (3 Cick ( Zouaoui and Wilson (b Zouaoui and Wilson (a See Henderson (3 for furter discussion Te input odel uncertainty proble aps to te setting in tis paper as follows Te rando object Z corresponds to a selection of input distributions and associated paraeters for a siulation experient Te rando variable X represents an estiate of a perforance easure fro te siulation odel Its distribution is dependent on te coice Z of input distributions and paraeters Te conditional expectation E(X Z represents te expected value of te perforance easure as a function of te input distributions and paraeters It is essentially wat one would copute fro te siulation if te siulation were allowed to run for an infinite aount of tie Notice tat it is still a rando variable owing to te uncertainty in te input distributions and paraeters A density of E(X Z gives a sense of te uncertainty in te estiate of te perforance easure due to te uncertainty in te values of te input distributions and paraeters Exaple 3 of Henderson (3 discusses tis density in te setting of a queueing siulation and describes ow it ay be interpreted Very little work as been done on te estiation of te distribution of a conditional expectation Te ost closely related work to ours involves te estiation of te distribution function of te conditional expectation E(X Z Lee and Glynn (999 considered te case were Z is a discrete rando variable Tis work was an outgrowt of Capter of Lee (998 were te case were Z is continuous is also considered We prefer to directly estiate te density because we believe tat te density is ore easily interpreted (visually tan a distribution function We use kernel density estiation etods to estiate te required

2 density Te analysis of our estiator draws fro etods tat are used in variable-bandwidt kernel density estiation etods (Hall 99 Andradóttir and Glynn (3 discuss a certain estiation proble tat in our setting is essentially te estiation of EX Teir proble is coplicated by te fact tat tey explicitly allow for bias in te estiator Suc bias can arise in steady-state siulation experients for exaple Tis paper is organized as follows In we describe our estiation etodology and sow tat under fairly general conditions te error in our density estiator converges at rate c /7 werec is te overall coputational budget We ten present soe nuerical exaples in 3 Soe brief conclusions and directions for future researc appear in 4 ESTIMATION METHODOLOGY Our proble is very siilar in structure to tat of Lee and Glynn (999 Accordingly we adopt uc of teir proble structure and assuptions in wat follows We assue te ability to draw saples fro te distribution P(Z and for any z in te range of Z to draw saples fro te conditional distribution P(X Z = z Let f denote te (target density of E(X Z wic we assue exists Let (Z i : i n be a sequence of independent identically distributed (iid copies of te rando variable Z Conditional on (Z i : i n te saple (X j (Z i : i n j consists of independent rando variables in wic X j (Z i follows te distribution P(X Z = Z i For ease of notation define µ( E(X Z = and σ ( Var(X Z = Suppose Y is a rando variable wit an unknown density g and (Y i : i n is a sequence of iid copies of te rando variable Y Te standard kernel density estiator at te value x is of te for ĝ(x; = n n i= ( x Yi were K is typically cosen to be a uniodal probability density function (pdf tat is syetric about zero and te sooting paraeter often referred to as te bandwidt is a positive nuber (Wand and Jones 995 p Te estiator can be crudely described as te su of equally weigted kernels centered at eac realization Y i If te kernel is a pdf te kernel spreads out te ass of /n syetrically about te neigborood of Y i In te case tat K is te pdf of a standard noral rando variable is te standard deviation and tus gives te spread of te kernels Tis estiator iediately suggests tat we can estiate f (x te density of E(X Z evaluated at x by were ˆ f (x; n = n X (Z i = n i= ( x X (Z i X j (Z i for i = n j= Note tat te values X (Z i at wic te kernels are centered are not realizations of te rando variable E(X Z as in te standard kernel density estiation setting described above but rater estiates We wis to analyze te asyptotics of te estiator For a given coputer budget c let = (c and n = n(c be cosen so tat te total coputational effort required to generate f ˆ(x; n is approxiately c Following Lee and Glynn (999 te coputational effort required to copute f ˆ(x; n is α n(c + α n(c(c were α and α are te average coputational effort used to generate Z i and X j (Z i respectively We need (c as c to ensure tat X (c (Z i E(X Z = Z i and we can assue α = witout loss of generality It follows tat (c and n(c ust be cosen to satisfy te asyptotic relationsip (cn(c/c asc Te bandwidt = (c is also a function of c To keep te notation less cubersoe te dependence of n and on c will be suppressed in te calculations Te error criterion tat we coose to use in analyzing te convergence is ean integrated squared error (ise defined as ise( ˆ f ( ; n = E ( ˆ f (x; n f (x dx Tis error criterion is not witout its drawbacks (see Devroye and Györfi 985 but its ateatical siplicity is appealing Switcing te order of integration yields ise( f ˆ( ; n = [ E ( f ˆ(x; n f (x ] dx Note tat te integrand is ean squared error (MSE wic decoposes into squared bias and variance We tus ave ise( ˆ f ( ; n = bias ( f ˆ(x; n dx + Var( f ˆ(x; n dx

3 In order to siplify te bias and variance calculations we ake te following additional assuptions Let N(a a denote a norally distributed rando variable wit ean a and variance a A Conditional on (Z i : i n X (Z i N(µ(Z i σ (Z i for i = n and are (conditionally independent; A Te kernel K is te density of a N( rando variable; A3 Te function µ( is strictly onotonic on its doain; A4 Te bandwidt is defined by = a δ were δ>anda > are constants independent of c If te central liit teore olds ten for large assuption A is approxiately true Tis assuption will be furter exained in 3 In A we specify te kernel K to be noral Togeter te norality of A and A allow us to derive copact expressions for bias and variance for te estiator f ˆ(x; n Tis will be illustrated in te proof presented in Assuption A3 is a siplifying assuption tat ensures tat a cange of variable tat we eploy later is valid Since we ave as c A4 ensures tat asc wic is necessary for convergence in te standard kernel density estiation setting Also note tat given A4 is copletely deterined by and δ Now te density estiator is a function of x n and δ so f ˆ(x; n and ise( f ˆ( ; n can now be written wit an abuse of notation as f ˆ(x; nδ and ise( f ˆ( ; nδ respectively In te case in wic te variance function σ ( is constant is exained In te general case is considered Constant Variance Function σ ( Te analysis for te case in wic te variance function σ ( is constant is siilar to tat in te standard kernel density estiation setting Hence te following assuptions togeter wit A and A4 are siilar to tose found in Prakasa Rao (983 p 44: A5 f is a bounded continuous square-integrable function; A6 n and as c Finally te constant variance assuption is noted: A7 σ ( σ > In Propositions and we ake use of o ( sall o" notation For sequences of real nubers a n and b n wesay tat a n = o(b n as n iff li a n/b n = n Taylor s Teore wit integral reainder is useful in te proof of Proposition We state it ere as a lea A proof can be found on p 78 of Apostol (967 Lea Assue f is twice continuously differentiable Ten f (x + = f (x + f (x + ( t f (x + tdt were and Proposition ise( ˆ f ( ; nδ = b = 4 Assue A-A7 Ten ( + σ b + b n ( +o ( + + n f (x dx b = π ( Since = (a/ /δ ( is equivalent to te following: for <δ ise( f ˆ( ; nδ= bc + b ( n + o + n and for δ> ( ise( f ˆ a 4/δ ( ; nδ= b + b ( n +o 4/δ + n were Proof: b c = 4 (ai(δ = + σ f (x dx E( f ˆ(x; nδ ( ( x = E X (Z ( [ ( ] x = E E X (Z Z ( Conditional on Z X (Z N(µ(Z σ

4 Since K is te density of a N( rando variable K ( / is te density of a N( rando variable Ten te conditional expectation above is a convolution of te densities of two noral rando variables and so E( f ˆ(x; nδ= E K + σ Define η = + σ so tat E ˆ f (x; n can be expressed as x µ(z + σ ( ( x E η K µ(z (3 η A cange of variable using A3 sows tat (3 is given by η K ( x y and anoter cange of variable gives f (x uηk (u du η f (y dy (4 Since f is twice continuously differentiable we ave by Lea tat E f ˆ(x; n = f (x K (u du η f (x uk(u du + η ( t f (x tuηu K (u dtdu = f (x + η ( t f (x tuηu K (u dtdu Ten we ave bias ( f ˆ(x; nδdx [ = η 4 ( t f (x tuηu K (u dtdu] dx A rater involved arguent tat uses Lebesgue s doinated convergence teore (see eg p 45 of Prakasa Rao 983 establises tat [ ( t f (x tuηu K (u dtdu] dx b as c It follows tat bias ( f ˆ(x; nδdx = ( + σ ( ( b + o + (5 Siilarly Var( f ˆ(x; nδ = [ ( ( ] x var n X (Z = [ ( ( ( x E n X (Z ( ( ( ] x E X (Z (6 = ( [ ( ] n π E x E / K X (Z / Z ( +o (7 n To obtain (7 we ave used te fact tat te second ter in (6 is f (x plus error ters as sown in te bias calculations above Terefore te second ter in (6 is of order n and terefore o((n We ten use te fact tat wen K ( ; is te density of a noral rando variable wit ean and variance K ( ; = K ( ; / π to obtain (7 Applying te sae steps to ( [ ( ] x E E / K X (Z / Z as were applied to ( [ E E ( ] x X (Z Z gives var( f ˆ(x; nδ= ( n π f (x + o n Ten (8 var( f ˆ(x; nδdx = ( n π + o (9 n and ( follows fro (5 and (9

5 Te norality given by A and A assures tat te convolutions in ( and (8 ave well defined fors Specifically for eac convolution we get a noral density evaluated at x wit expectation and variance resulting fro te su of te convolved distributions expectations and variances respectively Consider (4 wic is iediate fro te convolution perfored on te expectation Tis sows tat we are effectively doing standard kernel density estiation using kernels tat ave squared bandwidt η = + σ rater tan just Note tat te extra coponent σ is te variance of te error of te estiate X (Z about E(X Z conditional on Z So under te te stated assuptions te effect of using estiates X (Z of realizations of E(X Z rater tan actual realizations of E(X Z is an effective bandwidt wose square is wider tan by te variance of te error of te estiate Copare ( to te ise in standard kernel density estiation (Wand and Jones 995 ise(ĝ( ; = 4 b u K (u du + n K (u du + o( 4 + n ( Note tat wen K is te density of a N( rando variable u K (u du = and K (u du = π in wic case ( is precisely te sae as ( except for te order of te leading ter on te bias dx ter For ( it is 4 wereas for ( it is ( + σ Recalling tat = (a/ /δ weseeteorderonte bias dx ter in ( is actually larger for <δ<ascopared to ( For δ te order is te sae So te wider bandwidt arising fro using estiates rater tan realizations of E(X Z translates into diinised convergence on te bias dx ter in ise for <δ< We wis to coose n andδ to optiize te rate of convergence of ise( f ˆ( ; nδin ters of te coputer budget c and copare it to te optial rate of convergence for standard kernel density estiation To do tis we drop te low order ters and consider te large saple approxiation of ise( f ˆ( ; nδ: wic in turn give te optial rate of convergence of ise( f ˆ( ; nδ As entioned above we ust satisfy te asyptotic relationsip (cn(c/c asc For te large saple approxiation take n = c/ It turns out tat for any given δ we can solve for and tus n = c/ : { d (δc (δ = δ/(3δ+ if <δ d (δc δ/(δ+5 if δ> were ( b d (δ = a /(3δ+ c δ δ/(3δ+ b (δ + and ( δ/(δ+5 d (δ = a 5/(δ+5 4b b (δ + We note tat and n are suc tat assuption A6 olds for any δ Substituting into our expressions for aise( f ˆ( ; nδ gives { aise( f ˆ( ; ; n d ; δ = (δc δ/(3δ+ if <δ d (δc 4/(δ+5 if δ> were and d (δ = a /(3δ+ b δ/(3δ+ ( ( 3δ + b c δ (δ+/(3δ+ δ (δ + ( ( d 5 + δ (δ+/(δ+5 (δ = (ab 4/(δ+5 4b 4 δ + Tus te best rate of convergence is attained at δ = Ten te optial coice of is = dc /7 were ( 4b d = a /7 c /7 3b and te optial aise is aise( ˆ f ( ; nδ= { b c + b n if <δ if δ> ( a 4/δ b + b n Here aise stands for asyptotic ean integrated squared error We now solve for te n and δ tat give te optial rate of convergence of aise( f ˆ( ; nδ were aise( ˆ f ( ; ; n ; δ = dc 4/7 d = a /7 b 4/7 ( 7 4 ( 4b c 3/7 3

6 Te optial rate of convergence of ise is tus c 4/7 Altoug given above is te optial coice of for tis rate of convergence it is possible to acieve te rate c 4/7 so long as asyptotically (c = d 3 c 7 for any positive constant d 3 and of course n = c In standard kernel density estiation te optial rate of convergence is c 4/5 (Wand and Jones 995 p 3 Te decrease in rate of convergence is expected in tat for eac of te n observations X (Z i units of coputer tie are required and as c wereas in te standard kernel density estiation setting eac observation requires only one unit of coputer tie In addition as we noted above for <δ< te convergence of bias dx in te expression for ise is slower in tis setting as copared to standard kernel density estiation Te slower convergence for <δ< induces coosing δ = so tat it does play a role in deterining te optial rate of convergence General Variance Function σ ( In tis section assuption A7 is relaxed Define te function ρ( σ (µ ( We ake te following additional assuptions: A8 Te function ρ( is bounded above and also away fro zero is twice differentiable and its derivatives are continuous and bounded; A9 f f and f are square integrable Proposition Assue A-A6 A8 and A9 Ten for <δ ise( f ˆ( ; nδ = bv + b ( n + o + n and for δ> ( ise( f ˆ a 4/δ ( ; nδ= b + b ( n + o were 4 δ b v = [ f (x(ai(δ = + ρ(x 4 + f (xρ (x + f (xρ (x] dx + n and b and b are te sae as above Te proof of Proposition wic uses tecniques fro te variable bandwidt literature (Hall 99 but is oterwise siilar to te proof of Proposition will be given elsewere We reark tat te condition tat ρ( be bounded away fro zero is used only for te case δ ( Te only difference in ise in te constant variance function case and te general variance function case is in te coefficient of te bias dx ter wen <δ In te general case we ave b v wereas in te constant variance function case we ave b c We first note tat wen we ave a constant variance function (σ ( σ ρ( σ and ρ and ρ are zero so tat b v siplifies to bc Secondly we note tat in te general case σ ( plays a significant role in te constant b v troug te functions ρ ρ and ρ And finally we note tat if we define te function β( f ( ρ( b v is very siilar to (β (x dx 4 So we expect b v to be large wen te function β ( tends to be large in agnitude Following te sae line of reasoning as in te constant variance case te best rate of convergence is attained at δ = and = d v c /7 were ( 4b d v = a /7 v /7 3b Te optial aise is were aise( ˆ f ( ; ; n ; δ = d v c 4/7 d v = a /7 b 4/7 ( 7 4 ( 4b v 3/7 3 So te sae rate is acieved as in te constant variance case but te coefficient is different We again note tat te optial rate c 4/7 is attainable provided tat asyptotically = d 4 c /7 for any positive constant d 4 and n = c 3 EXAMPLES In tis section we exaine te convergence of a few basic exaples and copare to te teoretical results presented in Specifically for eac exaple we look at ise as a function of te coputer budget c For clarity we no longer suppress te dependence of our functions on c For exaple we now write ise(c (c andn(c

7 To estiate ise(c we first replicate te density estiator 5 independent ties: { ˆ f k ( ; (c n(c δ : k = 5} We define integrated squared error (ise as follows: ise(c = [ f ˆ(x; (c n(c δ f (x] dx log(mise log(mise vs log(c For eac k= 5 we use nuerical integration to approxiately copute ise k (c = Our estiate for ise(c is ten [ ˆ f k (x; (c n(c δ f (x] dx 5 5 ise k (c k= In calculating ˆ f k ( ; (c n(c δ we take δ = and (c = rc /7 as suggested in were te constant r was cosen in brief preliinary experients to be 3 for Exaple and for te oter exaples We took = /δ (so tat a = We estiate ise(c for te following values of c: {c = 4 l : l = 8} Exaple : In te first exaple we let Z Beta(4 4 (a Beta(a a rando variable as density on ( proportional to x a ( x a and conditional on Z = z X N(z 5 Ten te true density of te conditional expectation f is just te density of te Beta(4 4 distribution In Figure we plot log(ise(c vs log(c Te linearity of te plot suggests tat asyptotically ise(c = Vc γ for soe constants V and γ Teoretically we expect γ = 4/7 57 Note tat γ is te slope of te (log(c log(ise(c plot and te estiated slope of te plot in Figure is -54 Tis is very close to te expected rate of convergence Exaple : In tis exaple we consider a non-constant variance function σ ( Once again let Z Beta(4 4 Conditional on Z = z wetakex N(z z Te target density f is again te density of te Beta(4 4 distribution We present te log(ise(c vs log(c plot in Figure Te plot is linear and te slope is estiated to be -45 indicating poorer convergence as copared to Exaple Tis is likely te result of te variance function σ (z = z on te interval ( and zero elsewere We will furter discuss te ipact of tis variance function in Exaple log(c Figure : Mean Integrated Squared Error as a Function of Coputational Budget for Exaple but we note ere tat tis variance function does not satisfy te sootness assuption A8 at z = log(mise log(mise vs log(c log(c Figure : Mean Integrated Squared Error as a Function of Coputational Budget for Exaple Exaple 3: In tis exaple we study te ipact of violating te assuption A in wic we assue tat conditional on (Z i : i n X (Z i is norally distributed for i = n WetakeZ to ave a Beta(4 4 distribution sifted to te rigt by one unit so te support of Z is te interval ( Conditional on Z = z we now suppose X exp(/z ie conditional on Z = z X is exponentially distributed wit ean /z Note tat conditional on (Z i : i n X (Z i Gaa( Z i / for i = n (a Gaa(a a rando variable as density on ( proportional to x a e x/a so tat assuption A is violated Te target density f is te Beta(4 4 density sifted to te rigt by one unit

8 In Figure 3 we give te log(ise(c vs log(c plot Te slope is estiated to be -44 Te rates of convergence in Exaples and 3 are quite siilar suggesting tat te norality of X (Z i i = n is not crucial to te rate of convergence Tis is to be expected since te central liit teore (CLT tells us tat conditional on (Z i : i n for large X (Z i beaves approxiately like a rando variable wit a N(Z i Zi / distribution as in Exaple log(mise 5 5 log(mise vs log(c c = 644 For c = 644 te density estiate is sligtly skewed to te left Tis sae skewness was evident in oter independent replications of te experient We believe tat te skewness is a natural result of te for of te variance function σ (z = z on te interval ( and zero elsewere Recall tat to generate te estiated density our observations X (Z i i = n are sooted by noral kernels wit bandwidt ( + σ (Z i / / For large X (Z i Z i Ten te observations X (Z i i = n tat are larger in value are sooted ore tan te observations wit saller value resulting in te skewness seen in te plot Tis nonunifor sooting will probably be typical in exaples wit nonconstant variance but te teory presented in tis paper sows tat te convergence rate will not be affected Of course wile te rate ay not be affected te agnitude of te error ay be significantly affected troug ultiplicative constants density log(c Figure 3: Mean Integrated Squared Error as a Function of Coputational Budget for Exaple 3 c=48 c=644 true density x Figure 4: Te True Density and Two Estiators for Exaple 3 Te rate 44 is not as good as tat seen in Exaple We suspect tat tis reduction in rate and tat seen in Exaple is due to te function ρ( being discontinuous at z = Tis boundary effect is peraps evident in Figure 4 were te perforance of te density estiates deteriorates near te boundaries of te plot Te density estiate for c = 48 is quite poor but iproves wen 4 CONCLUSIONS AND FUTURE RESEARCH We ave sown ow to sare a coputational budget between external sapling of Z and internal sapling conditional on values of Z so as to iniize te aise of te density estiator Te aise can converge to at rate c 4/7 were c is te coputational budget Tis is slower tan te c 4/5 rate exibited in te standard density estiation context and bot of tese rates are slower tan te standard Monte Carlo rate c wen one is estiating an expectation Neverteless we believe tat te insigt one obtains fro te estiated density justifies te additional coputational effort involved Furterore one does not need an especially accurate estiate of te density in order to get soe idea of te extent of te effect of input uncertainty Clearly uc reains to be done We need to generalize our results beyond te case were Z is real-valued so as to capture ultiple input paraeters and/or distributions Te rate of convergence of te estiator sees to strongly depend on te sootness of ρ and as observed in experients not reported ere sootness of te target density We need to understand tis better In view of te relatively slow convergence of our estiators confidence intervals for estiates of f (x or ore generally confidence bands for te entire density f would be of great value A key assuption is tat X (z is exactly norally distributed Tis assuption often olds approxiately due to te central liit teore since we require to grow wit te coputational budget Te results for Exaple 3 suggest tat non-norality ay not severely ipact te rate of convergence but we need to better under-

9 stand tis ipact It is also of interest to consider probles tat do not fit te fraework ere suc as steady-state siulation and quantile estiation We ave sown ow to coose te bandwidt only up to a ultiplicative constant Tis constant can ave a strong ipact on te perforance of te estiator even toug it doesn t cange te asyptotic rate of convergence So just as in te standard kernel-density estiation case bandwidt selection reains an issue In view of te popularity of istogra estiators it would be interesting to explore teir asyptotic perforance in our setting Tey are known to converge at a slower rate tan kernel-based estiators in te iid setting (Freedan and Diaconis 98 We are pursuing or plan to pursue all of tese topics ACKNOWLEDGMENTS Te first autor was supported by a National Defense Science and Engineering Graduate Fellowsip Te work of te second autor was partially supported by National Science Foundation grants DMI 4884 and DMI 358 REFERENCES Andradóttir S and P W Glynn 3 Coputing Bayesian eans using siulation Subitted for publication Apostol T M 967 Calculus Volue I nd ed New York: Wiley Ceng R C H 994 Selecting input odels In Proceedings of te 994 Winter Siulation Conference ed J D Tew S Manivannan D A Sadowski and A F Seila 84 9 Piscataway NJ: IEEE Ceng R C H and W Holland 997 Sensitivity of coputer siulation experients to errors in input data Journal of Statistical Coputation and Siulation 57:9 4 Ceng R C H and W Holland 998 Two-point etods for assessing variability in siulation output Journal of Statistical Coputation and Siulation 6:83 5 Ceng R C H and W Holland 3 Calculation of confidence intervals for siulation output Subitted for publication Cick S E Input distribution selection for siulation experients: accounting for input uncertainty Operations Researc 49: Devroye L and L Györfi 985 Nonparaetric Density Estiation : Te L View New York: Wiley Freedan D and P Diaconis 98 On te istogra as a density estiator: L teory Z Warsceinlickeitsteorie verw Gebeite 57: Hall P 99 On te bias of variable bandwidt curve estiators Bioetrika 77 (3: Henderson S G 3 Input odel uncertainty: wy do we care and wat sould we do about it? In Proceedings of te 3 Winter Siulation Conference edse Cick P J Sáncez D J Morrice and D Ferrin To appear Piscataway NJ: IEEE Lee S H 998 Monte Carlo Coputation of Conditional Expectation Quantiles PD tesis Stanford University Stanford CA Lee S H and P W Glynn 999 Coputing te distribution function of a conditional expectation via Monte Carlo: discrete conditioning spaces In Proceedings of te 999 Winter Siulation Conference ed P A Farrington H Black Nebard D T Sturrock and G W Evans Piscataway NJ: IEEE Prakasa Rao B L S 983 Nonparaetric Functional Estiation New York: Acadeic Press Wand M and M Jones 995 Kernel Sooting London: Capan & Hall Zouaoui F and J R Wilson a Accounting for input odel and paraeter uncertainty in siulation In Proceedings of te Winter Siulation Conferenceed BAPetersJSSitDJMedeirosandMW Rorer 9 99 Piscataway NJ: IEEE Zouaoui F and J R Wilson b Accounting for paraeter uncertainty in siulation input odeling In Proceedings of te Winter Siulation Conference ed B A Peters J S Sit D J Medeiros and M W Rorer Piscataway NJ: IEEE AUTHOR BIOGRAPHIES SAMUEL G STECKLEY is a PD candidate in te Scool of Operations Researc and Industrial Engineering at Cornell University His priary field of interest is input odel uncertainty in discrete-event siulation He is te recipient of an NDSEG fellowsip His e-ail address is <steckley@oriecornelledu> SHANE G HENDERSON is an assistant professor in te Scool of Operations Researc and Industrial Engineering at Cornell University He as previously eld positions at te University of Micigan and te University of Auckland He is an associate editor for te ACM Transactions on Modeling and Coputer Siulation Operations Researc Letters and Mateatics of Operations Researc andtenewsletter editor for te INFORMS College on Siulation His researc interests include discrete-event siulation queueing teory and sceduling probles His e-ail address is <sg9@cornelledu> and is web page URL is <wwworiecornelledu/ sane>