VARIANCE ESTIMATION USING REPLICATED BATCH MEANS. Sigrún Andradóttir Nilay Tanık Argon

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1 Proceedings of the 2001 Winter Siulation Conference B.A.Peters,J.S.Sith,D.J.Medeiros,andM.W.Rohrer,eds. VARIANCE ESTIMATION USING REPLICATED BATCH MEANS Sigrún Andradóttir Nilay Tanık Argon School of Industrial and Systes Engineering Georgia Institute of Technology Atlanta, GA , U.S.A. ABSTRACT We present a new ethod for obtaining confidence intervals in steady-state siulation. In our replicated batch eans ethod, we do a sall nuber of independent replications to estiate the steady-state ean of the underlying stochastic process. In order to obtain a variance estiator, we further group the observations fro these replications into nonoverlapping batches. We show that for large saple sizes, the new variance estiator is less biased than the batch eans variance estiator, the variances of the two variance estiators are approxiately equal, and the new steady-state ean estiator has a saller variance than the batch eans estiator when there is positive serial correlation between the observations. For sall saple sizes, we copare our replicated batch eans ethod with the (standard) batch eans and ultiple replications ethods epirically, and show that the best overall coverage of confidence intervals is obtained by the replicated batch eans ethod with a sall nuber of replications. 1 INTRODUCTION The proble of constructing a confidence interval on the steady-state ean µ of a stationary process {X i } i 1 often arises in siulations. This requires a point estiator for µ and also an estiator for the variance of that point estiator. The usual point estiator for µ is the saple ean, X n = n i=1 X i /n, wheren is the total nuber of observations. The ore challenging part of constructing a confidence interval is estiating the variance of the saple ean, or equivalently, σ 2 n = nvar( X n ). If the nuber of observations n is large, then the experienter can instead estiate the variance paraeter, σ 2 = li n σ 2 n. There is a significant aount of research on the proble of obtaining a good estiator for σ 2 in the discreteevent siulation literature (see, e.g., Alexopoulos and Seila, 1998). Two of the ost popular ethods for estiating σ 2 are the ethod of batch eans (BM) and the ethod 338 of ultiple independent replications (MR). With the BM ethod, the observations obtained fro one long run are grouped into non-overlapping batches to estiate σ 2. With the MR ethod, on the other hand, σ 2 is estiated based on the replication eans obtained fro the independent replications (which usually start in the sae initial conditions). We provide a ore detailed background on these ethods in Section 2. The trade-off between aking a single long run and any independent replications is studied by any authors in the siulation literature (see, e.g., Kelton and Law, 1984, Whitt, 1991, and the references therein). Most of these studies provide supporting evidence in favor of aking one long run (which is the case for the BM ethod). This is in part due to the fact that the MR ethod has ore data that is containated by initialization bias than the BM ethod (due to using initial conditions that do not represent the long-run behavior of the underlying stochastic process). The ost failiar ethod for dealing with the initialization bias is to truncate soe of the initial observations fro each replication. This iplies that the MR ethod wastes ore data than the BM ethod. However, the MR ethod also has soe benefits. One advantage of the MR ethod is that the replication eans are independent. For the BM ethod, on the other hand, the batch eans are usually positively correlated. Another advantage of using ultiple replications is that there is a chance to start the individual replications in different initial states and observe various different saple paths of the underlying stochastic process. This is especially useful for stochastic processes such as nearly decoposable Markov chains. In such situations, if the single run of the BM ethod is not long enough, the experienter ay not detect any abnoralities in the process. With several independent replications, on the other hand, there is a chance that the special structure of the Markov chain ight be caught. In this study, our objective is to introduce a new confidence interval estiation ethod that cobines the advantages of the BM and MR ethods. For this purpose, we

2 suggest collecting a total of n observations in k > 1 independent replications. We assue that k issallsothatwe cannot siply use the replication eans to obtain a variance estiator. Therefore, after collecting the data in k independent replications, we further partition the observations in each replication (possibly after accounting for initialization bias by truncating soe of the initial observations) into b > 1 batches, each consisting of observations. Hence, the total nuber of batches will be kb. We refer to this ethod as the replicated batch eans (RBM) ethod. Since our RBM ethod involves doing ore than one replication, it has fewer correlated batches than the BM ethod and has the potential to observe different saple paths of the underlying stochastic process that start in different initial conditions. Moreover, since the RBM ethod uses a saller nuber of replications than the MR ethod, its steady-state ean estiator is likely to be less biased than the steady-state ean estiator obtained by the MR ethod. Also, with the RBM ethod, less data will be deleted (due to initialization bias) than with the MR ethod. This paper is organized as follows. In Section 2, we provide background aterial on the BM and MR ethods. In Section 3, we introduce the RBM ethod and present our asyptotic results for the estiators for µ and σ 2.We copare these results to the corresponding results for the BM ethod in Section 4. In Section 5, we present nuerical results for the RBM, BM, and MR ethods applied to a single exaple. Finally, we give our concluding rearks in Section 6. 2 BACKGROUND 2.1 Batch Means (BM) In Sections 2, 3, and 4 of this paper, we assue that the underlying stochastic process {X i } i 1 is stationary. The original BM ethod akes a single long run and then partitions the observations X 1,...,X n fro this run into kb > 1 non-overlapping batches, each consisting of observations (we assue that n = kb). Then, X j = 1 i=1 X ( j 1)+i is the j th batch ean, where j = 1,...,kb, X BM = ni=1 X i /n is the point estiator for the steady-state ean µ, and V BM = kb 1 kb j=1 ( X j X BM ) 2 is the BM estiator for σ 2 n = nvar( X BM ). When the nuber of batches kb is fixed and is large, we can assue Andradóttir and Argon 339 that the batch eans are approxiately i.i.d. (independent and identically distributed) noral rando variables (see, e.g., Glynn and Iglehart 1990). Therefore, an approxiate 100(1 α)% confidence interval for µ is given by µ X BM ± t kb 1,α/2 V BM /n, where t d,α/2 is the 1 α/2 quantile of the t distribution with d degrees of freedo. 2.2 Multiple Replications (MR) For the ultiple independent replications ethod, kb independent replications of the underlying stochastic process, each of length observations, are perfored to estiate µ and σ 2. We define X (r) i to be the i th observation fro the r th replication for i = 1,..., and r = 1,...,kb. We also let Ȳ r = 1 i=1 X (r) i be the r th replication ean, where r = 1,...,kb. This yields the point estiator X MR = kb r=1 Ȳr /kb for µ. A point estiator for σ 2 can be obtained by the saple variance, V MR = kb 1 kb r=1 (Ȳr X MR ) 2. Finally, note that when is large, we can assue that Ȳ 1,...,Ȳ kb are approxiately i.i.d. noral rando variables. Therefore, an approxiate 100(1 α)% confidence interval for µ is given by µ X MR ± t kb 1,α/2 V MR /n. 3 REPLICATED BATCH MEANS (RBM) As we entioned earlier, the RBM ethod involves running a sall nuber k of independent replications of the underlying stochastic process and grouping the observations in each replication into b non-overlapping batches of size. Let X (r) j = 1 i=1 X (r) ( j 1)+i be the j th batch ean of the r th replication, where r = 1,...,k and j = 1,...,b, X RBM = k b r=1 j=1 X (r) j /n be the point estiator for µ, and

3 Andradóttir and Argon V RBM = kb 1 k b r=1 j=1 ( X (r) j X RBM ) 2 be the RBM estiator for σ 2. For stationary processes, we can assue that the batch eans obtained fro each replication are approxiately i.i.d. noral rando variables (the batch eans fro different replications are always independent). With this assuption, we obtain the following approxiate 100(1 α)% confidence interval for µ, µ X RBM ± t kb 1,α/2 V RBM /n (for a rigorous proof and explicit conditions, we refer the reader to Andradóttir and Argon 2001). Before we state our results on the asyptotic expectation and variance of X RBM and V RBM for a stationary process {X i } i 1, we define the covariance function R j = Cov(X 1, X 1+ j ), j = 1, 2,..., and the relevant quantity γ = 2 j=1 jr j. We also use the notation g(n) = O( f (n)) to express that g(n)/f (n) C for soe constant C and all n 1, and g(n) = o( f (n)) to indicate that g(n)/f (n) 0asn. We first study the properties of the point estiator, X RBM. Since the stochastic process under consideration is assuedtobestationary, X RBM is an unbiased estiator for the steady-state ean (i.e., E[ X RBM ]=µ). Thus, the ean squared error of the estiator, MSE( X RBM ), equals Var( X RBM ). The next result gives an expression for the variance of X RBM. Theore 1. Under ild oent and ixing conditions (see Andradóttir and Argon 2001), we have Var ( ) σ X 2 RBM = kb + γ kb o 2. We next present our results on the expected value, variance, and ean squared error of the RBM variance estiator, V RBM,forlarge. Theore 2. Under the sae conditions as in Theore 1, we have E [ V ( ] kb RBM = σ ) ( ) γ 1 + (kb 1) b + o. Theore 2 shows that, as, V RBM is an asyptotically unbiased estiator for σ 2, regardless of the choice of k and b. Theore 3. Under slightly ore restrictive oent and ixing conditions than the ones needed to prove Theores 1 and 2 (see Andradóttir and Argon 2001), we have kbvar ( V RBM ) = as and kb. 2kb(kb + 1) σ 4 (kb 1) 2 + O kb + O 1/4 = 2σ 4 + o (1), (1) Fro Theores 2 and 3, we have MSE ( V ) γ 2 RBM = 2 + 2σ 4 ( 1 kb + o 2 ) + o. (2) kb Hence, for large and kb, the estiator V RBM is consistent in ean square. 4 BM vs. RBM In this section, we copare our results obtained in Section 3 with the corresponding results for the BM ethod. First, we note that by the stationarity of the underlying stochastic process, both X BM and X RBM are unbiased estiators for the steady-state ean (i.e., E[ X RBM ]=E[ X BM ]=µ). Therefore, coparing the variances of X BM and X RBM is equivalent to coparing their ean squared errors. Fro Song and Scheiser (1995) and Theore 1 above, we get Var ( X BM ) Var ( X RBM ) = (1 k)γ k 2 b o 2. Thus, if γ < 0(e.g.,ifR j 0forall j {1, 2,...}, corresponding to a stochastic process with positive serial correlation), then the RBM ethod offers a point estiator with a saller variance than the point estiator obtained by the BM ethod for large. This is a notable result, since positive correlation occurs frequently in siulations (for exaple, the waiting tie processes in various queueing systes exhibit positive correlation). We now turn our attention to the confidence intervals for µ obtained by the BM and RBM ethods. The halflength of a generic 100(1 α)% confidence interval is defined by H = t d,1 α/2 ( V /n) 1/2,whered is the nuber of degrees of freedo, V is the variance estiator, and n is the total nuber of observations. Aong confidence interval estiation ethods achieving coverage of approxiately 1 α/2, the one with the sallest E[H ] is preferred, and then that with the sallest Var(H ). For both the BM and RBM ethods, we have d = kb 1andn = kb. Hence, the only ter that is different in the half-lengths 340

4 of the confidence intervals obtained by the BM and RBM ethods is the variance estiator. Therefore, a coparison of the variance estiators obtained by the two ethods for large will also yield insights about the half-length estiators in the resulting confidence intervals. We conclude this section by coparing the BM and RBM ethods in ters of expected value, variance, and ean squared error of the variance estiator for large. By Theore 1 of Chien, Goldsan, and Melaed (1997) and Theore 2 above, we get Bias ( V BM ) Bias ( V RBM ) = γ (k 1) kb(kb 1) + o, which is positive for large. Hence, the RBM ethod offers an asyptotically less biased variance estiator than the BM ethod. Consequently, the RBM ethod can be expected to provide an asyptotically less biased half-length estiator than the BM ethod. This is iportant, since a less biased half-length helps in achieving the desired coverage of the corresponding confidence interval. Finally, under the conditions of Theore 3, Chien, Goldsan, and Melaed (1997) show that for large and kb, Var( V BM ) and MSE( V BM ) are equal to the right-hand sides of equations (1) and (2), respectively. Thus, we can conclude that for a stationary process and large and kb, the BM and RBM ethods will yield variance estiators with approxiately the sae variance and ean squared error. 5 NUMERICAL RESULTS In this section, we study the sall-saple behavior of the RBM, BM, and MR ethods. For this purpose, we consider the first-order autoregressive process with exponential arginals, EAR(1). In particular, for all i 1, we have X i = { φxi 1 with probability φ, φx i 1 + ɛ i with probability 1 φ, where 0 φ<1and{ɛ i } i 1 are i.i.d. exponential rando variables with rate one (see Lewis, 1980). Our priary criterion for coparing the RBM, BM, and MR ethods is the coverage P{µ X ± H } of the resulting confidence intervals, where X is the point estiator for µ and H is the half-length of the confidence interval (we say that a good coverage is achieved when P{µ X ± H } is very close to 1 α, the desired confidence level). We have siulated the EAR(1) process for φ {0.1, 0.5, 0.9} and exained the coverages and the expected half-lengths of the confidence intervals under the BM, RBM, and MR ethods. For each ethod, we let = 4and kb = 32. Consequently, the BM ethod has 32 batches, the RBM ethod with k {2, 4, 8, 16} replications has Andradóttir and Argon /k batches per replication, and the MR ethod has 32 replications with one batch per replication (we have not truncated any initial observations). In Tables 1, 2, and 3, we provide the estiated coverage values when X 0 is distributed uniforly between 0 and a {0.00, 0.25, 0.50,...,5.00}. Noting that the stationary distribution is exponential with ean one, it is clear that we have a relatively good choice of the initial distribution when a is close to two, and that when a is close to zero or five, then we have a poor choice of the initial distribution. All the nuerical results that are presented in this section are based on 10,000 independent acro replications. For each acro replication, we have used coon rando nubers to generate observations fro the EAR(1) process for the three ethods. In the last two rows of Tables 1, 2, and 3, we also provide the averages and the standard deviations of the coverages over all initial distributions for each ethod. Table 1: Coverages of Confidence Intervals for the EAR(1) Process with φ = 0.1 and1 α = 0.90 a BM RBM MR avg std First, we note that for all values of φ that we have considered in this study, the BM, RBM, and MR ethods all yield under-coverage (except for the MR ethod with a good choice of the initial distribution and φ = 0.1), which is partially due to the sall saple size and the positive correlation inherent in the EAR(1) process. For φ = 0.1, the coverages of the BM, RBM, and MR ethods for each value of a are very siilar, as can be seen fro Table 1. This is an expected result, since with a sall value of φ, the EAR(1) process behaves alost like an i.i.d. stochastic process. In Tables 2 and 3, for each choice of a, we have highlighted the coverages of the ethod with the best coverage.

5 Andradóttir and Argon Table 2: Coverages of Confidence Intervals for the EAR(1) Process with φ = 0.5 and1 α = 0.90 a BM RBM MR avg std Table 3: Coverages of Confidence Intervals for the EAR(1) Process with φ = 0.9 and1 α = 0.90 a BM RBM MR avg std When we observe each row of Tables 2 and 3, we can see that our RBM ethod perfors better than both the BM and MR ethods for reasonably good choices of the initial distribution. This suggests that the RBM ethod is ore than an average of the BM and MR ethods. More specifically, the RBM ethod shows the best perforance for φ = 0.5 when a {0.50,...,1.50, 2.75,...,5.00} and it is the best for φ = 0.9 whena {0.75,...,1.50, 2.50,...,4.75}. Moreover, the RBM ethod never perfors worse than both the BM and MR ethods. As one ight expect, the MR ethod perfors better than the other two ethods when the initial distribution is quite good (i.e., when a {1.75, 2.00, 2.25, 2.50} for φ = 0.5 and a {1.75, 2.00, 2.25} for φ = 0.9). On the other hand, in this exaple the BM ethod provides the best coverage for very poor choices of initial distribution (i.e., when a {0.00, 0.25} for φ = 0.5 and when a {0.00, 0.25, 0.50, 5.00} for φ = 0.9). Fro the last row of Tables 2 and 3, we can see that the BM ethod and the RBM ethod with a sall nuber of replications provide saller standard deviations of the coverage than the other ethods. This suggests that the BM ethod and the RBM ethod with a sall nuber of replications are not very sensitive to the choice of the initial distribution, whereas the MR ethod and the RBM ethod with a large nuber of replications show perforance that depends heavily on the choice of the initial distribution. For exaple, depending on the choice of the initial distribution, the MR ethod yields both the best and the worst results aong all the ethods. We have also observed the perforance of the BM, RBM, and MR ethods starting at various initial distributions in ters of the resulting half-lengths. Our conclusion is that there is not uch of a difference aong the BM, RBM, and MR ethods in this respect. To illustrate this point, we provide the average half-lengths of the confidence intervals over all the initial distributions considered in Tables 1, 2, and 3 in Table 4. Table 4: Average Half-lengths of Confidence Intervals for the EAR(1) Process with 1 α = 0.90 φ BM RBM MR CONCLUSIONS For φ = 0.5 andφ = 0.9, it can be observed that the RBM ethod with a sall nuber of replications k, is overall the best in ters of the average coverage (recall that our initial intention was to apply the RBM ethod with a sall nuber of replications). Moreover, the overall perforance of the BM ethod is better than that of the MR ethod. 342 We have presented a new confidence interval estiation ethod for steady-state siulation. In our replicated batch eans (RBM) ethod, we do a sall nuber of independent replications to estiate the steady-state ean and then group the observations that are collected fro these replica-

6 Andradóttir and Argon tions into non-overlapping batches to estiate the variance paraeter. We have shown that for a stationary process and large batch sizes, the RBM ethod provides a less biased variance estiator than the batch eans (BM) ethod, regardless of the choice of the nuber of batches and replications. Moreover, we have shown that for a stationary process, the variances of the two variance estiators becoe approxiately equal as both the nuber of batches and batch sizes grow. Finally, we have shown that when there is positive serial correlation in the stationary process of interest, then the point estiator obtained by the RBM ethod has a saller variance than the point estiator obtained by the BM ethod for large batch sizes. This is iportant, since ost real-life queueing systes produce positively correlated data. We have also copared the perforance of the BM, RBM, and MR ethods for sall saple sizes, using a non-stationary stochastic process. The nuerical results suggest that our RBM ethod with a sall nuber of replications provides better coverages than both the BM and MR ethods for reasonably good choices of the initial distribution. The MR ethod perfors the best only for good choices of the initial distribution, and, in our exaple, the BM ethod appears to have the best coverage for very poor choices of the initial distribution, especially when the observations have high positive autocorrelation. REFERENCES Alexopoulos, C., and A. F. Seila Output Data Analysis. Chapter 7 in the Handbook of Siulation Principles, Methodology, Advances, Applications, and Practice, ed. J. Banks, New York: John Wiley and Sons, Inc. Andradóttir, S., and N. T. Argon Replicated batch eans for steady-state siulation. Working paper. Chien, C. H., D. Goldsan, and B. Melaed Largesaple results for batch eans. Manageent Science 43 (9): Glynn, P. W., and D. L. Iglehart Siulation output analysis using standardized tie series. Matheatics of Operations Research 15 (1): Kelton, W. D., and A. M. Law An analytical evaluation of alternative strategies in steady-state siulation. Operations Research 32 (1): Lewis, P. A. W Siple odels for positive-valued and discrete-valued tie series with ARMA correlation structure. In Multivariate Analysis V: Proceedings of the Fifth International Syposiu on Multivariate Analysis, ed. P. R. Krishnaiah, New York: North-Holland. Song, W.-M. T., and B. W. Scheiser Optial eansquared-error batch sizes. Manageent Science 41 (1): Whitt, W The efficiency of one long run versus independent replications in steady-state siulation. Manageent Science 37 (6): ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grants DMI and DMI AUTHOR BIOGRAPHIES SIGRÚN ANDRADÓTTIR is an Associate Professor in the School of Industrial and Systes Engineering at the Georgia Institute of Technology. Previously, she was an Assistant Professor and later an Associate Professor in the Departents of Industrial Engineering, Matheatics, and Coputer Sciences at the University of Wisconsin Madison. She received a B.S. in Matheatics fro the University of Iceland in 1986, an M.S. in Statistics fro Stanford University in 1989, and a Ph.D. in Operations Research fro Stanford University in Her research interests include siulation, applied probability, and stochastic optiization. She is a eber of INFORMS and served as Editor of the Proceedings of the 1997 Winter Siulation Conference. NİLAY TANIK ARGON is a Ph.D. student in the School of Industrial and Systes Engineering at the Georgia Institute of Technology. She received her B.S. and M.S. degrees in Industrial Engineering fro the Middle East Technical University, Ankara, Turkey, in 1996 and 1998, respectively. In August 2000, she received an M.S. degree in Operations Research fro the Georgia Institute of Technology. Her priary research interests are the odeling and analysis of stochastic systes and siulation output analysis. She is a eber of INFORMS. Her e-ail address is nilay@isye.gatech.edu. 343