Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence New Jersey Insttute of Technology Newark, New Jersey, 070, USA ABSTRACT We consder the problem of estmatng the tme-average varance constant for a statonary process A prevous paper descrbed an approach based on multple ntegratons of the smulaton output path, and descrbed the effcency mprovement that can result compared wth the method of batch means whch s a specal case of the method In ths paper we descrbe versons of the method that have low bas for moderate smulaton run lengths The method constructs an estmator based on applyng a quadratc functon to the smulaton output The partcular quadratc form s chosen to mnmze varance subject to constrants on the order of the bas Estmators that are frst-order and second-order unbased are descrbed INTRODUCTION We consder the steady-state smulaton output analyss problem for a statonary process Suppose a smulaton produces output Y n = Y,Y,,Y n Our goal s to estmate n σ = lm n Var Y /n n = One approach s to apply a quadratc functon to the output vector Y n In general, the cost of computng such an estmator s proportonal to the square of the smulaton run length n An example of ths approach s the batch means estmator, though n the case of batch means the complexty s lnear n run length n due to the specal form of the quadratc functon Our approach s as follows Choose a nonnegatve parameter k that s small compared wth the square root of the smulaton run length n As the smulaton runs, mantan a vector W n R k+ after n smulated steps At any tme n we can construct our varance estmator V n from W n n tme Ok by applyng a quadratc functon to W n Ths way the complexty s reduced to Ok = On per computaton of the varance estmator A prevous paper Calvn 007, ntroduced a class of estmators that are based on multple ntegratons of the smulated path; the rth component of the vector W n s the r-fold ntegrated path of the smulaton output In that paper, the quadratc functon was chosen so that the lmtng dstrbuton of the estmators could be worked out easly and we were manly concerned wth the effcency of the estmators, the effcency of an estmator s defned as the recprocal of the product of mean-squared error and computaton tme Numercal experments showed that wth the partcular choce of quadratc functon used there, the effcency of the resultng estmator could be an order of magntude hgher than the effcency of the batch means estmator However, n experments wth short run lengths t was found that the bas of the estmators was qute large compared wth the method of batch means That fndng motvated the present paper, whch focuses on bas propertes SETUP AND ASSUMPTIONS Suppose that a smulaton generates a real-valued sequence Y,Y, We assume that Y s a statonary process wth EY = µ and that the seres σ = EY µ + E Y µy n µ n> converges absolutely wth σ > 0 For the bas approxmatons, we wll assume the absolute convergence of the sums for j λ j = n j E Y µy n µ n> -444-6-0/07/$00 007 IEEE 3
3 SIMULATION ALGORITHM We now outlne the proposed method for estmatng the parameter σ of the smulated process Choose a parameter k 0 the ntegraton count parameter Run a smulaton, producng output {Y,Y,} Defne W j 0 = 0 for 0 j k and for > 0 set and for j k set W j = Y l l= W 0 = j l= W j l The data mantaned by the smulaton method after n smulated steps s W n = W 0 n, W n,, W k n We wll use bold typeface for vectors and matrces The vector W n can be updated n tme Ok at each smulaton step Defne nt Wn 0 t = n / Y, 0 t, and for r and 0 t, Wnt r = Wn r sds = r Wn 0 st s r ds = Set W n t = Wn 0 t,,wn k t and W n t = W n 0 t,, W n t k, 0 t Defne the matrces A and N n by { r q [ r Ar,q = q], q r, 0 q > r, the [ r j] are the Strlng numbers of the frst knd Knuth 997, and N n r,q = { n q+/, q = r, 0 q r The followng theorem provdes the means to transform the dscrete terated sums nto the terated ntegrals Theorem W n = AN n W n In order to center the output, defne W 0 ns = W 0 n s sw 0 n, 0 s, and for r and 0 t, W r nt = r W 0 nst s r ds Set W n t = W 0 nt,,w k nt, 0 t Defne Then 0 0 / 0 0 /3 0 0 R = /k+ 0 0 W n = I RAN n W n Our output analyss method s based on the followng lmt theorem Theorem As n, W n,,w k n D σn 0,C, C = C j s the matrx defned by C j = j,, j k + j+ + j+ The proof of Theorem follows from Theorem n Calvn 007, wth a dfferent covarance matrx due to a slghtly dfferent defnton of the W r n 4 BIAS AND VARIANCE To prepare for the constructon of low bas estmators, we wll need an approxmaton to the expectaton of products of the {W r n} The followng theorem approxmates the expectaton to order n Theorem 3 For α,β k, E W α n W β αβσ n = α + β + α + β + + Dα,β + Dα,β n n + o n 4
as n, and D α,β αβ σ = α + β + αβ + λ α + β +, 3 D α,β = σ α + β Iα= + + λ αi α> β + + λ αα +ββ α + β α β + I α> β α + I β> + I β= + βi β> α + α + β αi α> β + βi β> α + and = j= then β = + j j + j I > j+ ji j> B j = 0, + 7 EW T n BW n = σ + β n + β n + o n, k = k j= + j j+ I > j + I j> B j We now consder estmators of the form V n = W T n BW n for a symmetrc postve semdefnte matrx B of order k We wll consder dfferent matrces B, chosen to optmze dfferent objectves In general, we wll choose B to mnmze the asymptotc varance of V n, subject to constrants on the bas We next gve the bas expanson n terms of the matrx B Theorem 4 Suppose that the matrx B s chosen so that =0 j=0 j + j+ + j+ B j = 4 If n addton we choose B such that then = j= j+ + j+ B j = 0, EW T n BW n = σ + β + o n, n β = = j= j + j+ B j If, n addton to 4 and, the followng constrants hold: = j= I= + I > j+ + I j= + ji j> + + j B j = 0 6 Suppose that the matrx B satsfes 4 and, and defne Then from Theorem 4, V n = WT n BW +β /n 8 E V n σ = o n, so Vn s a frst-order unbased estmator In general, there could be many choces of B that satsfy 4 and, thus gvng rse to many frst-order unbased estmators Among these we want to dentfy one wth mnmal varance Let us approxmate the varance of the estmator usng the lmtng dstrbuton gven n Theorem ; that s, Var W T n BW n trcbcb, whch s a convex functon of B here tr denotes the trace of a matrx Thus we are led to a convex optmzaton problem: We want to choose B to mnmze trcbcb subject to the constrants 4 and The frst-order optmalty condtons are a set of k + lnear equatons Thus we can fnd the matrx B that mnmzes the asymptotc varance subject to the bas constrants by solvng a system of k + lnear equatons We let Vn denote the estmator defned by 8 wth B chosen to solve the constraned optmzaton problem The constructon of a second-order unbased estmator s analogous to the constructon we gave for the frst-order unbased estmator In ths case we add the constrants 6 and 7 If B satsfes these addtonal constrants and we
defne V n = then Theorem 4 mples that W T n BW +β /n+β /n, E V n σ = o n ; that s, Vn s a second-order unbased estmator Choosng B to mnmze the asymptotc varance now requres solvng a system of k + 4 lnear equatons One can also choose B to satsfy only 4 and mnmze the asymptotc varance We denote ths estmator by Vn 0 OVERLAPPING BATCHES Gven a smulaton output of length n, Y,Y,,Y n, and a batch sze b, 0 < b < n, we can construct estmators based on the smulaton output Y j+,y j+,,y j+b for 0 j n b and then average the estmators Ths approach has been used n the constructon of several varance constant estmators; see Meketon and Schmeser 984 and Alexopoulos et al 006 For 0 < b < n and 0 j n b consder the estmator for the jth batch of the form V j,n = W T j, j+b B W j, j+b, W j, j+b = W j, 0 j+b,, W j, k j+b, and for r, W j, 0 j+b j+b = Y, = j+ W r j, j+b = r n = j+ W r m, The followng Lemma shows how to obtan the vectors W j, j+b from the vectors W 0, j+b = W j+b produced by the basc smulaton algorthm descrbed n Secton 3 Lemma W j, j+b = W j+b D W j, 9 Dr,q = { r r q b r q, q r, 0, q > r and a r s the rsng factoral functon defned by a r = aa+a+ a+r and a 0 = ; see Knuth 997 Now W j, j+b = I RAN b W j, j+b s the analog of the vector W n on whch we based the estmators n Secton 3, but defned for the jth batch of sze b Then T V j,n = W j+b D W j Bb W j+b D W j, B b = I RN b A T B I RN b A Our overlapped estmator s then V ov n = n b n b+ j=0 V j,n By choosng B as descrbed n Secton 4 we can obtan frst- or second-order unbased estmators We need to replace n wth b, so for example the frst-order unbased estmator has bas ob as b 6 NUMERICAL EXPERIMENTS The numercal results are for a frst-order autoregressve process defned by Y = ϕy + ε,, wth the ε N0, ndependent and Y 0 N0, We set the parameter ϕ = 09, whch results n σ = 9 In all cases we constructed the overlappng estmators based on a batch sze of b = n/0 Fgures,, and 3 show the results of experments for run lengths of,000, 6,000, and 0, 000, respectvely The curves for each of the mth order unbased estmators start at the upper left for k = and as k ncreases the bas generally ncreases whle the varance decreases For each run length, the fgures show the sample bas and sample varance of the standard overlappng batch means estmator Alexopoulos et al 006, and the frst and second order unbased estmators The choce of k ranged from to 4 Each experment conssted of 0 ndependent replcatons 6
40 3 0 0 0 0 0 3 4 6 7 8 0 0 0 03 04 0 06 07 Fgure : Sample bas and varance for AR smulatons, varyng k, n =,000 Fgure 3: Sample bas and varance for AR smulatons, varyng k, n = 0,000 3 0 0 <http://webnjtedu/ calvn/pepdf> [accessed June 0, 007] Knuth, D 997 The art of computer programmng, volume : Fundamental algorthms thrd ed Readng, Massachusetts: Addson-Wesley Meketon, M S, and B W Schmeser 984 Overlappng batch means: somethng for nothng? In Proceedngs of the 984 Wnter Smulaton Conference, 7 Pscataway, New Jersey: Insttute of Electrcal and Electroncs Engneers 0 0 04 06 08 4 6 Fgure : Sample bas and varance for AR smulatons, varyng k, n = 6,000 As the run length n ncreases, the bas of the nd order unbased estmator becomes very small compared to that of the batch means estmator Experments wth the consstent estmator V 0 n resulted, as expected, n large bas and low varance, and were omtted from the fgures AUTHOR BIOGRAPHY JAMES M CALVIN s an assocate professor n the Department of Computer Scence at the New Jersey Insttute of Technology He receved a PhD n operatons research from Stanford Unversty and s an assocate edtor for ACM Transactons on Modelng and Computer Smulaton and INFORMS Journal on Computng In addton to smulaton output analyss, hs research nterests nclude global optmzaton and probablstc analyss of algorthms REFERENCES Alexopoulos, C, N Argon, D Goldsman, N Steger, G Tokol, and J Wlson 006 Effcent computaton of overlappng varance estmators for smulaton To appear n INFORMS Journal on Computng Calvn, J M 007 Smulaton output analyss usng ntegrated paths To appear n ACM Transactons on Modelng and Computer SImulaton; avalable va 7