Skart: A skewness- and autoregression-adjusted batch-means procedure for simulation analysis

Transcription

1 Forthcoming in IIE Transactions. Skart: A skewness- and autoregression-adjusted batch-means procedure for simulation analysis ALI TAFAZZOLI 1 Metron Aviation, Inc., Catalina Ct. Suite 101, Dulles, VA 20166, USA JAMES R. WILSON 2 Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Campus Box 7906, Raleigh, NC , USA 1 Member, Institute of Industrial Engineers. tafazzoli@metronaviation.com : Telephone: (703) Fax: (703) Fellow, Institute of Industrial Engineers, and corresponding author. jwilson@ncsu.edu : Telephone: (919) Fax: (919) skart-iiet28.tex June 20, :32

2 Skart: A skewness- and autoregression-adjusted batch-means procedure for simulation analysis Dedicated to the Memory of Richard E. Rosenthal Skart is an automated sequential batch-means procedure for constructing a skewness- and autoregressionadjusted confidence interval (CI) for the steady-state mean of a simulation output process either in discrete time (i.e., using observation-based statistics), or in continuous time (i.e., using time-persistent statistics). Skart delivers a CI designed to satisfy user-specified requirements concerning both the CI s coverage probability and its absolute or relative precision. Skart exploits separate adjustments to the classical batch-means CI to account for the effects on the distribution of the underlying Student s t-statistic arising from skewness and autocorrelation of the batch means. The skewness adjustment is based on a Cornish-Fisher expansion for the classical batch-means t-statistic, and the autocorrelation adjustment is based on a first-order autoregressive approximation to the batch-means autocorrelation function. Skart also delivers a point estimator for the steady-state mean that is approximately free of initialization bias. The associated warm-up period is based on iteratively applying von Neumann s randomness test to spaced batch means with increasing sizes for each batch and its preceding spacer. In extensive experimentation, Skart compared favorably with its competitors. Supplementary materials are available for this article. Go to the publisher s online edition of IIE Transactions for additional discussion, detailed proofs, etc. Keywords: Simulation, statistical analysis, steady-state analysis, method of batch means, Cornish-Fisher expansion, autoregressive representation 1. Introduction Most simulations can be classified as either finite-horizon or steady-state. A finite-horizon (terminating) simulation usually has a given starting condition, and it ends at a specific time or with the occurrence of a specific terminating condition. On the other hand, a steady-state (nonterminating) simulation operates (at least conceptually) into the indefinite future; and in this case interest centers on long-run average performance, which presumably is independent of the starting condition. In a steady-state simulation experiment, the user typically seeks to construct point and confidence-interval (CI) estimates for some parameter or characteristic of the steady-state cumulative distribution function (c.d.f.) of a discrete- or continuous-time output process. In this article, interest is focused primarily on estimating the steady-state mean X of a selected discrete-time univariate process fx i W i D 1;2;:::g. (The Online Supplement to this article explains how to adapt the methods presented here for estimating the steady-state mean of a continuous-time process.) skart-iiet28.tex 1 June 20, :32

3 Three fundamental problems arise in analyzing the output from a steady-state simulation. The start-up (or initialization bias) problem is caused by a transient in the initial sequence of responses that is due to the simulation s starting condition. In practical applications, it is usually impossible to start a simulation in steady-state operation; instead users often do the following: (a) start the simulation in some fixed, convenient initial condition that may not be typical of steady-state operation; and (b) select the duration of the warm-up period (data-truncation point, statistics-clearing time) so that beyond the warm-up period, the mean of each simulation-generated observation is sufficiently close to the steady-state mean. If observations prior to the end of the warm-up period are included in the analysis, then the resulting estimator may be biased (Law, 2007); and such bias in the point estimator can severely degrade not only the accuracy of the point estimator but also the probability that the associated CI will cover the steady-state mean. In addition to the start-up problem, the user is often faced with the correlation problem, which is caused by pronounced stochastic dependence between successive responses generated within a single simulation run. Moreover, the user is often also faced with the nonnormality problem, which is caused by pronounced departures from normality, especially nonzero skewness, in successive responses generated within a single simulation run. The three foregoing problems complicate the construction of a CI for the steady-state mean because standard statistical methods require independent and identically distributed (i.i.d.) normal observations to yield a valid CI; and thus other analysis techniques are needed. Several methods have been devised for steady-state simulation analysis. The method of nonoverlapping batch means (NBM) is the basis for the procedure of Law and Carson (1979), the ABATCH and LBATCH procedures (Fishman and Yarberry, 1997), and ASAP (Steiger and Wilson, 2002). The NBM method divides the sequence of simulation-generated outputs into adjacent nonoverlapping batches of sufficiently large size so that the resulting batch means are approximately i.i.d. and normal with expected value X ; and thus a CI for X can be computed using the classical Student s t-ratio. ASAP3 (Steiger et al., 2005) and SBatch (Lada et al., 2008) are two recent batch-means procedures in which the size of the warm-up period and the size of all subsequent batches are taken separately to be just large enough to yield adjacent nonoverlapping batch means (in the case of ASAP3) or spaced batch means (in the case of SBatch) that approximately constitute a stationary first-order autoregressive (AR(1)) process and thus are normal with the AR(1) autocorrelation structure. Suitable modifications to the half-length of the classical batch-means CI for X are then based on the following: (i) in the case of ASAP3, a Cornish-Fisher expansion for the associated Student s t-statistic; and (ii) in the case of SBatch, a simple correlation adjustment to the sample variance of the spaced batch means. WASSP (Lada and Wilson, 2006) and the Heidelberger-Welch procedure (Heidelberger and Welch, 1981) seek to deliver valid CIs for X by estimating the power spectrum of the output process generated by a steady-state simulation model. This article documents Skart, a new sequential procedure for steady-state simulation output analysis that is an extension of the classical NBM method. The name Skart is an abbreviation of the phrase Skewnessand autoregression-adjusted Student s t analysis ; moreover, skart (or scart) is an English word with the now-obsolete meaning to gather together carefully (Oxford English Dictionary, 1989). Skart addresses the start-up problem by iteratively applying the randomness test of von Neumann (1941) to spaced batch skart-iiet28.tex 2 June 20, :32

4 means with increasing sizes for each batch and its preceding spacer. Skart addresses the nonnormality problem by exploiting a Cornish-Fisher expansion for the classical Student s t-ratio based on a random sample from a nonnormal (skewed) distribution; and analysis of this expansion leads to a modified t-ratio that incorporates terms due to Johnson (1978) andwillink (2005) so as to compensate for any skewness in the final set of truncated, nonspaced batch means. Skart addresses the correlation problem by using a first-order autoregressive model of the truncated, nonspaced batch means to compensate for correlation between those batch means. To achieve the user-specified precision in the final CI for X, Skart may request additional simulation-generated observations; and several iterations of Skart may be performed until a CI satisfying the precision requirement is finally delivered. The remainder of the article is organized as follows. Section 2 provides the necessary background information on the NBM method for steady-state simulation analysis and summarizes the key problems with using the NBM method in practice. Section 3 contains a description of Skart, including a high-level overview of its structure and operation together with a formal algorithmic statement of the procedure. Section 4 contains derivations of the skewness and correlation adjustments to the classical NBM Student s t-ratio that form the basis for Skart. Section 5 summarizes selected results from the experimental performance evaluation of Skart. Finally Section 6 recapitulates the main findings of this research and provides recommendations for future work. The Online Supplement to this article contains not only a detailed explanation of each step in the operation of Skart but also complete proofs of the main theoretical results used in the formulation of the procedure. Tafazzoli et al. (2008) is a preliminary, abridged version of this article. Tafazzoli et al. (2010b) summarizes the results of an extensive performance evaluation of Skart and some of its competitors on a wide range of test problems. Tafazzoli et al. (2010a) documents the design and evaluation of N-Skart, a nonsequential variant of Skart. See Tafazzoli (2009) for a comprehensive discussion of the design, justification, and experimental evaluation of Skart and N-Skart. 2. Background and Problem Statement In the NBM method, the sequence of simulation-generated outputs fx i W i D 1;:::;ng is divided into k adjacent nonoverlapping batches, each of size m, wherem is sufficiently large to ensure that the resulting batch means are at least approximately i.i.d. normal random variables with expected value X (Law, 2007). The sample mean of the j th batch is Y j.m/ D 1 mx X.j 1/mCi for j D 1;:::;kI (1) m id1 and the grand average of the individual batch means, xy D xy.m;k/d 1 k kx Y j.m/ ; (2) is used as a point estimator for X. The objective is to construct a CI estimator for X that is centered on a point estimator of X like xy in (2), where in practice some initial observations (or batches) may be deleted skart-iiet28.tex 3 June 20, :32 j D1

5 (truncated) to eliminate the effects of initialization bias. We assume that when the simulation is in steady-state operation (that is, warmed up), the simulationgenerated process fx i g is stationary (in the strict sense) so that the joint distribution of the fx i g is insensitive to time shifts. In steady-state operation, we also assume the process fx i g is weakly dependent that is, X i s widely separated from each other in the sequence are almost independent (in sense of -mixing; see Billingsley (1968)) so that the lag-q covariance, X.q/ D EŒ.X icq X /.X i X / for q D 0; 1; 2;:::; tends to zero sufficiently fast as jqj!1. These weakly dependent processes typically obey a central limit theorem of the form p h i n X.n/ x D X! N 0; n!1 X ; (3) where xx.n/ D n 1 P n id1 X i is the sample mean for a time series of length n; X D lim nvar xx.n/ 1X D n!1 X.q/ (4) qd 1 is the variance parameter (as distinguished from the marginal variance X 2 D VarŒX i D X.0/); and the D symbol! denotes convergence in distribution. See, for example, Theorem 1 in Section 4.2 below for n!1 general conditions under which the right-hand side of (4) is absolutely convergent so that X is well defined and the asymptotic property (3) holds. Although some output analysis methods exploit a strongly consistent estimator of the variance parameter X in constructing a CI for X, the classical NBM method is based on a different approach. A key assumption of the NBM method is that the batch size m is sufficiently large so that the batch means fy j.m/ W j D 1;:::;kg are i.i.d. normal variates, where the symbol i.i.d. fy j.m/ W j D 1;:::;kg i.i.d. N X ;ƒ m =m ; (5) is read is independent and identically distributed as ; and ƒ m D m VarŒY j.m/ D X.0/ C 2 m 1 X `D1 1 ` m X.`/ : It follows that lim ƒ m D X and VarŒY j.m/ X =m, provided m is sufficiently large. If (5) holds exactly, then we can apply classical results concerning Student s t-distribution to compute a CI for X.In particular if (5) holds exactly, then from the sample variance of the batch means, S 2 m;k D 1 k 1 kx Yj.m/ xy.m;k/ 2 ; j D1 we obtain Sm;kı 2 k as an unbiased estimator of Var Y.m;k/ x that is independent of xy.m;k/and has a scaled chi-squared distribution with k 1 degrees of freedom; and thus the classical NBM Student s t-statistic, Y.m;k/ t D x q X ı ; k S 2 m;k skart-iiet28.tex 4 June 20, :32

6 has Student s t-distribution with k 1 degrees of freedom so that for a user-specified confidence coefficient 1 where 0< <1, an exact /%CIfor X is xy.m;k/ t 1 =2;k 1 S m;k p k ; (6) where t u; denotes the u quantile of Student s t-distribution with degrees of freedom. Moreover, under the same weak-dependence conditions that are sufficient to ensure (3) asm!1with k fixed, Equation (36) in Theorem 1 below implies that as m!1,(5) holds asymptotically so that (6) is an asymptotically valid CI for X. The main difficulty with conventional NBM procedures such as the procedure of Law and Carson (1979) and the ABATCH and LBATCH procedures of Fishman and Yarberry (1997) isthe lack of a reliable method to determine a sufficiently large batch size m so that the batch means fy j.m/g are approximately normal, uncorrelated, and free of initialization bias. For more this issue, see Steiger et al. (2005) andlada et al. (2006). 3. Description of Skart 3.1. High-level overview of Skart Skart is a sequential extension of the classical NBM method. Skart addresses the start-up problem by iteratively applying the randomness test of von Neumann (1941) to spaced batch means with increasing sizes for each batch and its preceding spacer. When the randomness test is finally passed with a batch size m and spacer size dmfor sufficiently large integers m and d (where m 1 and d 0), the data-truncation point (i.e., the length of the warm-up period) w is defined by the initial spacer so that the leading w D dm observations are truncated (ignored) in calculating the point and CI estimators of X. (Although the batch size m may be increased on subsequent iterations of the later steps of Skart in order to satisfy the precision requirement as explained below, the truncation point w remains fixed once the randomness test has been passed.) Beyond the data-truncation point w, Skart computes k 0 truncated, nonspaced batch means with batch size m, Y j.m/ D 1 mx X wc.j 1/mCi for j D 1;:::;k 0 I (7) m id1 and then Skart computes the sample mean and variance of the truncated, nonspaced batch means, xy.m;k 0 / D 1 Xk 0 k 0 Y j.m/ and S 2 m;k D 1 Xk 0 0 Yj k 0.m/ xy.m;k 0 / 2 ; (8) 1 j D1 respectively. Next Skart computes an asymptotically valid /% skewness- and autoregressionadjusted CI for X having the form 2 s s 3 4 xy.m;k 0 / G.L/ AS 2 m;k 0 k 0 j D1 ; xy.m;k 0 / G.R/ AS 2 m;k 0 k 0 5 ; (9) skart-iiet28.tex 5 June 20, :32

7 where the skewness adjustments G.L/ and G.R/ are defined in terms of the function G./ D p 3 1 C 6ˇ. ˇ/ 1 2ˇ for all real ; with ˇ D y B m;k 00 6 p k 00 (10) and 8 9 ˆ< approximately unbiased estimator of the marginal skewness of Y j.m/ that is computed from k 00 spaced batch means with truncation point w, batch size m, and >= yb m;k 00 D ˆ: >; dw=mem ignored observations in each interbatch spacer, so that the skewness-adjustment function G./ has the arguments L D t 1 =2;k 00 1 and R D t =2;k 00 1 ;and the correlation adjustment A is computed as i A D h1 Cy' Y.m/ i.h1 y' Y.m/ ; (11) where the standard estimator of the lag-one correlation of the truncated, nonspaced batch means is y' Y.m/ D 1 k 0 1 X Yj.m/ xy.m;k 0 / Y j C1.m/ xy.m;k 0 / k 0 1 S 2 : (12) j D1 m;k 0 (Note that in Equation (10), the indicated cube root 3p 1 C 6ˇ. ˇ/ is understood to have the same sign as the quantity 1 C 6ˇ. ˇ/.) Therefore G.L/ and G.R/ are skewness-adjusted quantiles of Student s t-distribution for the left- and right-hand subintervals of the CI (9) with respect to the point estimator xy.m;k 0 /; and the autoregression (correlation) adjustment A is applied to the naive estimator S 2 ı m;k k 0 0 of Var xy.m;k 0 / so as to compensate for any residual correlation between the truncated, nonspaced batch means (7) that are used to compute the truncated grand average xy.m;k 0 /. The specific methods for computing w, m, k 0, k 00,andB y m;k 00 are detailed in the Online Supplement to this article. The final step of Skart is to determine whether the CI of the form (9) satisfies the user-specified precision requirement. The half-length of this CI is taken to be q H D maxf jg.l/j; jg.r/j g AS 2 ı m;k k 0 ; (13) 0 the maximum of the left- and right-hand subintervals of (9). If the CI (9) satisfies the precision requirement H H ; where H is given by 8 ˆ< 1 ; for no user-specified precision level ; H D r ˆ: ˇˇ xy.m;k 0 / ˇˇ ; for a user-specified relative precision level r ; h ; for a user-specified absolute precision level h ; then Skart terminates, delivering the CI (9) and the more conventional CI of the form xy.m;k 0 / H. If the precision requirement H H is not satisfied, then Skart estimates the total number of truncated, nonspaced batches of the current batch size m that are needed to satisfy the precision requirement, k D.H=H / 2 k 0 I and thus k m is the latest estimate of the total sample size beyond the truncation point that skart-iiet28.tex 6 June 20, :32

8 is needed to satisfy the precision requirement. For the next iteration of Skart, the batch count is taken to be k 0 D min k ; 1,024 ; and the associated batch size is updated according to ( ) m; if k 0 k ; m m mid 1:02; k =k 0 ;2:0 (14) ; if k 0 <k ; where midfu 1 ;u 2 ;u 3 g denotes the median of the numbers u 1, u 2,andu 3. (In Section A1.6 of the Online Supplement, we elaborate the reasons for (a) the upper limit of 1,024 on the batch count; and (b) the range.1:02;2:0/ for the multiplier by which the batch size is increased on a single iteration of Skart.) On the next iteration of Skart, the total sample size including the warm-up period is given by n D w C k 0 m. The additional observations are obtained by restarting the simulation or by retrieving extra data from storage; and then the next iteration of Skart involves reperforming the operations of Equations (7) (14) with the updated values of k 0, m,andn. Successive iterations of Skart are performed until a CI satisfying the precision requirement is finally delivered. Figure 1 depicts a high-level flowchart of Skart. An implementation of Skart in Visual Basic and the Skart User s Manual are available online via Start Collect observations, compute their sample skewness, and set batch size Compute nonspaced batch means and their sample skewness; set max batches per spacer Independence test passed? Yes Skip first spacer; reinflate batch count and reset batch size No Add another batch to each spacer; recompute spaced batch means No Reached max batches per spacer? Yes Compute nonspaced batch means and autoregressionadjusted variance estimator Compute spaced batch means and associated skewness-adjusted t-ratio Increase batch count or batch size; collect observations No Increase batch size; deflate batch count; set spacer size to zero and collect additional observations Compute skewnessand autoregressionadjusted CI CI meets precision requirement? Yes Deliver CI Fig. 1. High-level flow chart of Skart Formal algorithmic statement of Skart To invoke Skart the user must provide the following: (a) a desired CI coverage probability 1, where 0< <1; and (b) an upper bound H on the CI half-length H as defined by (13), where H is either expressed in absolute terms as the maximum acceptable value of H, or in relative terms as the maximum skart-iiet28.tex 7 June 20, :32

9 acceptable fraction r of the magnitude of the associated point estimator of X. Subsequently Skart delivers one of the following: (i) a nominal /% CIfor X that satisfies the specified absolute or relative precision requirement, provided no additional data are required; or (ii) a new, larger sample size n to be supplied to Skart when it is executed again. If additional observations must be generated by resuming (continuing) the current run of the user s simulation model before a CI with the required precision can be delivered, then Skart must be called again with the additional data; and this cycle may be repeated several times before Skart finally delivers a CI. A formal algorithmic statement of Skart is given in Figure 2. A detailed explanation of the individual steps of Skart is provided in the Online Supplement to this article. 4. Basic Results Underlying Skart Section summarizes the fundamental results underlying the skewness adjustments to Student s t-statistic as formulated by Johnson (1978)andWillink (2005), and Section describes the adaptation of these skewness adjustments to the batch-means t-statistic used in Skart. Section summarizes the fundamental results underlying the correlation adjustment based on autoregressive moving average time series models of batch means; and Section describes the adaptation of the correlation adjustment to the skewnessadjusted CI finally delivered by Skart. Complete proofs of all the results stated in Section are given in the Online Supplement to this article Skewness adjustment to the NBM Student s t-statistic Skewness adjustment to Student s t-statistic based on Johnson (1978) and Willink (2005) First we consider the situation in which the basic observations fx i W i D 1;:::;ng are i.i.d. with sample mean xx D n 1 P n id1 X i and sample variance S 2 D.n 1/ 1 P n id1.x i xx/ 2, where the c.d.f. F./ of the fx i g has finite moments of all orders; and in this subsection we use a simplified notation for the mean D EŒX i D R C1 1 x df.x/; the variance 2 D E.X i / 2 ; the central moments ` D E.X i /` for ` D 2;3;:::; and the associated cumulants ` for ` D 1;2;:::. The first four cumulants of F./ are given by 1 D ; 2 D 2 ; 3 D 3 ;and 4 D If we want to emphasize the dependence of the parameters, 2, `, ` on the random variable X, then we will write.x/, 2.X/, `.X/, `.X/. In using Student s ratio t D p n xx ı S to form CIs for when the fx i g are i.i.d., the main problem arises when the underlying c.d.f. F./ has pronounced positive or negative skewness so that the coverage probability of the usual CI based on Student s t-distribution is severely degraded. To avoid this problem, Johnson (1978) proposes a modification of Student s t-ratio based on a Cornish-Fisher expansion of this statistic involving the mean, variance, and higher central moments (or cumulants) of the underlying c.d.f. F./. The general form of a Cornish-Fisher expansion for the random variable X with moments, 2, 3, skart-iiet28.tex 8 June 20, :32

10 [1] From the initial sample fx i W i D 1;:::;ng of size n 1,280, compute the sample mean xx, variance S 2, and skewness B y of the last 1,024 observations as follows: xx ` 1 P n idn `C1 X i I S 2.` 1/ 1 P n idn `C1.X i xx / 2 I and B y f`=œ.` 1/.` 2/ g P n idn `C1.X i xx / 3 =S 3. If ˇˇ ybˇˇ >4:0, then set the initial batch size m 16 and increase the initial sample size to n 20,480; otherwise set m 1. Set the current number of batches in a spacer, d 0, andthe maximum number of batches allowed in a spacer, d 10. Divide the initial sample into k 1,280 nonspaced (adjacent) batches of size m; and compute the corresponding nonspaced batch means according to Equation (1). Set the randomness test size, ran 0:20, and the number of times the batch count has been deflated in the randomness test, b 0. [2] Compute the sample mean xy.m;`/, sample variance S 2 m;`, and sample skewness yb m of the last 80% of the current set of k nonspaced batch means fy j.m/ W j D 1;:::;kg as follows: ` b0:8kci xy.m;`/ ` 1 P k j Dk `C1 Y j.m/ I S 2 m;`.` P 1/ 1 k j Dk `C1 ŒY j.m/ xy.m;`/ 2 I and yb m f`=œ.` 1/.` 2/ g P k j Dk `C1 ŒY j.m/ xy.m;`/ 3 =S 3.Ifˇˇ y m;` Bmˇˇ >0:5, then reset the maximum number of batches per spacer, d 3. [3] Apply the von Neumann test for randomness to the current set of nonspaced batch means fy j.m/ W j D 1;:::;kg using the significance level ran. [3a] If the randomness test is passed, then set k 0 k andgoto[5]; otherwise go to [3b]. [3b] Insert spacers each with m observations (one ignored batch) between the k 0 bk=2c remaining nonignored batches; and set the total number of batches in a spacer, d 1. Assign the spaced batch means with batch size m and spacer size dm, Y j.m; d/ 1 m mx X fj.dc1/ 1gmCi for j D 1;:::;k 0 : (15) id1 [3c] Apply the randomness test to the current set of k 0 spaced batch means fy j.m; d/ W j D 1;:::;k 0 g using the significance level ran. If the randomness test is passed, then go to [5]; otherwise go to [3d]. [3d] If d D d so that the current number of batches per spacer has reached its maximum, then go to [4]; else add another ignored batch to each spacer so that the total number of batches per spacer and the number of spaced batches are respectively updated according to d d C 1 and k 0 n ı f.d C 1/mg : Reassign the spaced batch means fy j.m; d/ W j D 1;:::;k 0 g according to (15) and go to [3c]. [4] Update the batch size m, the total batch count k, the overall sample size n, and Skart s other status variables according to m p 2m ; k d0:9ke; n km; d 0; b b C 1; and d 10 : Obtain the required additional simulation-generated observations, recompute the k nonspaced batch means with batch size m according to Equation (1),andgoto[2]. Fig. 2. Algorithmic statement of Skart. skart-iiet28.tex 9 June 20, :32

11 [5] Skip the first w dm observations in the overall sample of size n. Divide the remaining n 0 n w observations into k 0 k0.1=0:9/ b nonspaced batches of size m maxfbn 0 =k 0 c;mg. If mk 0 >n 0, then obtain mk 0 n 0 additional observations. Compute the current set of truncated, nonspaced batch means fy j.m/ W j D 1;:::;k 0 g according to Y j.m/ 1 m mx X wc.j 1/mCi for j D 1;:::;k 0 : (16) id1 [6] Compute the grand average xy.m;k 0 / and the sample variance S 2 m;k 0 of the current set of k 0 truncated, nonspaced batch means fy j.m/ W j D 1;:::;k 0 g according to Equation (8). Use the sample estimator of the lag-one correlation of the truncated, nonspaced batch means, y' Y.m/ 1 k 0 1 kx 0 1 j D1 h ih i. Y j.m/ xy.m;k 0 / Y j C1.m/ xy.m;k 0 / S 2 m;k ; 0 to compute the correlation adjustment, h i.h i A 1 Cy' Y.m/ 1 y' Y.m/ ; that is to be applied to the naive estimator S 2 m;k 0 ı k 0 of Var xy.m;k 0 /. [7] Compute the skewness- and correlation-adjusted /%CIfor X using the modified critical values G.t 1 =2;k 00 1/ and G.t =2;k 00 1/ of Student s t-ratio for k 00 1 degrees of freedom, 2 4 xy.m;k 0 / G.t 1 =2;k 00 1/ s AS 2 m;k 0 k 0 ; xy.m;k 0 / G.t =2;k 00 1/ s AS 2 m;k 0 k ; (17) where to evaluate (17), spaced batch means are computed with truncation point w, batch size m, and d 0 dw=me batches per interbatch spacer so that there are k 00 1 Cb.k 0 1/=.d 0 C 1/c spaced batch means, Y j.m; d 0 / 1 m mx X wc.j 1/.d 0 C1/mCi for j D 1;:::;k 00 ; (18) id1 whose grand average, sample variance, and sample third central moment are respectively given by xy.m;k 00 ;d 0 1 Xk 00 / k 00 Y j.m; d 0 /; (19) j D1 S 2 m;k 00 ;d 0 1 X Yj k 00.m; d 0 / xy.m;k 00 ;d 0 / 2 ; (20) 1 k 00 j D1 Fig. 2 (continued). Algorithmic statement of Skart. skart-iiet28.tex 10 June 20, :32

12 k 00 X T m;k 00 ;d 0 Yj.k 00 1/.k 00.m; d 0 / xy.m;k 00 ;d 0 / 3 : (21) 2/ k 00 j D1 From the latter statistics (20)and(21), compute the sample skewness yb m;k 00 of the spaced batch means and the associated skewness-adjustment function G./ to be applied to the CI (17), yb m;k 00 T m;k 00 ;d 0 S 3 m;k 00 ;d 0 and G./ p 3 1 C 6ˇ. ˇ/ 1 2ˇ for all, whereˇ yb m;k 00 6 p : (22) k00 Compute the half-length of the current CI, H max jg.t 1 =2;k 00 1/j; jg.t =2;k 00 1/j s AS 2 m;k 0 k 0 : If no precision level is specified, then deliver the CI of the form (17) and the more conventional CI of the form xy.m;k 0 / H and stop; otherwise go to [8]. [8] Apply the appropriate absolute- or relative-precision stopping rule. [8a] If the half-length H of the current CI satisfies the user-specified precision requirement H H ; (23) where H ( r ˇˇ xy.m;k 0 /ˇˇ; for a user-specified relative precision level r ; h ; for a user-specified absolute precision level h ; ) (24) then deliver both the CI of the form (17) and the more conventional CI of the form xy.m;k 0 / H and stop; otherwise go to [8b]. [8b] Estimate the number of batches of the current size m required to satisfy (23) (24), k.h=h / 2 k 0 : [8c] Update the number of nonspaced batch means k 0, the batch size m, and the total sample size n as follows: k 0 min k ; 1,024 ; ( m; if k 0 k ; m m mid 1:02; k =k 0 ;2:0 ; if k 0 <k ; n w C k 0 m: Obtain the additional simulation-generated observations; recompute the truncated, nonspaced batch means fy j.m/ W j D 1;:::;k 0 g according to (16) and go to [6]. Fig. 2 (continued). Algorithmic statement of Skart. skart-iiet28.tex 11 June 20, :32

13 4 and associated cumulants 3, 4 based on a standard normal random variable Z is given by CF.X/ D C Z C Z2 1/ C Z3 Z/ C I (25) see, for example, Sections of Stuart and Ord (1994). For the sample mean xx D xx.n/ of a random sample of size n from F./, we have the moments. xx/ D ;.xx/ D p ; `. xx/ D ` for ` D 3;4;::: I (26) n n` 1 see, for example, Sections of Abramowitz and Stegun (1972). Combining (25) and(26), we see that a valid representation of xx by a Cornish-Fisher expansion paralleling (25) up to terms of order n 1 is CF. xx/ D C p Z C 3 n 6n 2.Z2 1/ C O P n 3=2 ; (27) because the first neglected term in (27) expressed as a function of the moments of the underlying c.d.f. F./, the sample size n, and the standard normal random variable Z is 4. xx/ xx/.z3 Z/ D 4 24 n 3=2.Z3 Z/ D n 3=2.Z3 Z/ D O P n 3=2 ; where in general for sequences of random variables fa n W n D 1;2;:::g and fb n W n D 1;2;:::g, the notation A n D O P.B n / means that given an arbitrarily small >0, there is a constant M D M./ and a positive integer n 0 D n 0./ such that Pr ja n jmjb n j 1 for every n>n 0 ; see Lehmann (1999, p. 54). In particular the notation A n D O P.n 3=2 / means that with high probability as n increases, A n tends to zero at least as fast as n 3=2. Johnson (1978) seeks an improved CI for by eliminating the low-order terms involving 3 in the Cornish-Fisher expansion of the modified t-statistic t 1 D. xx / C C. xx / 2. 2 =n/ p : (28) S 2 =n The Cornish-Fisher expansion for t 1 is obtained by inserting into (28) Cornish-Fisher expansions for xx and ıp S 2 and by exploiting the relation Corr. xx;s 2 / D /. The quantities and are chosen to yield the following properties for the Cornish-Fisher expansion of t 1 : (a) the constant terms (i.e., the terms that do not involve random variables) sum to zero; and (b) all terms that involve 3 and have the form O P.n 1=2 / are eliminated. Johnson (1978) shows that if the values D 3 =.3 4 / and D 3 =.2n 2 / are substituted into (28), then in the Cornish-Fisher expansion for t 1 any residual effects arising from skewness of the original observations fx i g have the form O P.n 1 / and thus become negligible as the sample size n increases. Making the indicated substitutions for and in (28) yields the modified t-statistic t 1 D. xx / C 3 =.6 2 n/ C Œ 3 =.3 4 /. xx / 2 p S 2 =n ; (29) skart-iiet28.tex 12 June 20, :32

14 which has approximately Student s t-distribution with n 1 degrees of freedom provided the fx i g are randomly sampled from a distribution whose moments of all orders exist, and n is sufficiently large. Unfortunately the numerator of t 1 is a quadratic function of the unknown quantity ; and when the sample estimates y Dy 3 =.3S 4 / and y Dy 3 =.2nS 2 / are used in (29), in general the resulting function of is not monotone and can fail to have a real-valued inverse. Equation (29) can also yield CIs for consisting of two disjoint subintervals, which is highly counterintuitive and therefore undesirable. Willink (2005) formulates a further modification of Johnson s statistic (29) by adding a cubic term in. xx / to the numerator of (29), ultimately yielding the modified t-statistic t 2 D. xx / Cy 3 =.6S 2 n/ C Œy 3 =.3S 4 /. xx / 2 C Œy 2 3 =.27S 8 /. xx / 3 p S 2 =n (30) that is a strictly monotone function of with an easily computed inverse, where y 3 D n=œ.n 1/.n 2/ P n id1.x i xx/ 3 is the usual unbiased sample estimator of 3. The additional term in the Cornish-Fisher expansion of t 2 has the form O P.n 1 /, which has the same order as the first neglected term in the Cornish- Fisher expansion of t 1. Thus, we do not expect the extra term in the Cornish-Fisher expansion of t 2 to have a large effect on the asymptotic performance of CIs based on t 2 as n!1. If we define the real-valued function Q.u/ D u C ˇ C 2ˇu 2 C 4 3ˇ2u 3 for u 2 R,where ˇ D y 3=S 3 6 p n ; (31) then it is easy to see that Q./ is strictly increasing with inverse 8 p 9 3 ˆ< 1 C 6ˇ. ˇ/ 1 G./ Q 1 ; if ˇ 6D 0; >=./ D 2ˇ for all 2 R : (32) ˆ: >; ; if ˇ D 0; It follows from (31) and(32) that we can express Willink s statistic (30) as a strictly increasing function of the usual Student s t-statistic, t 2 D Q p n xx ı S D G 1p n xx ı S I and if this transformation ensures that t 2 has approximately Student s t-distribution with n 1 degrees of freedom, then for each 2.0; 1/ an approximate /% two-sided CI for is given by xx G.t 1 =2;n 1 / p S ; xx G.t =2;n 1 / S p : (33) n n The performance evaluation of Willink (2005) shows that when the distribution of the fx i g is skewed, the CI (33) proposed by Willink appears to be considerably more reliable (in terms of approximately achieving the nominal coverage probability 1 ) than the usual CI based on the standard Student s t-statistic or other CIs in the literature. When the fx i g are i.i.d. normal, the CI (33) is slightly wider on average than the standard CI. In general, the coverage of Willink s CI (33) increases as the magnitude of the skewness skart-iiet28.tex 13 June 20, :32

15 of the fx i g decreases, the sample size n increases, or the nominal confidence coefficient 1 increases. Moreover, the results in Tables 1 3 of Willink (2005) show that when the sample size is large enough, (33) performs well for distributions with skewness as high as 4:0. For higher skewness values, Willink s CI (33) often fails to achieve an acceptable coverage probability Adaptation of the skewness adjustment to Skart In the context of applying N-Skart to correlated data aggregated into k 00 approximately i.i.d. spaced batch means with batch size m and spacer size d 0 m as defined by (18), a skewness-adjusted CI for the associated steady-state mean X of the form 2 s s 3 S 2 S 2 4 xy.m;k 00 ;d 0 m;k / G.t 1 =2;k 00 1/ 00 ;d 0 k 00 ; xy.m;k 00 ;d 0 m;k / G.t =2;k 00 1/ 00 ;d 0 5 k 00 (34) can be obtained by making the following substitutions in the definition (31) (32) of the function G./ and in the skewness-adjusted CI (33): (i) xx is replaced by xy.m;k 00 ;d 0 / asgivenin(19); (ii) S is replaced by S m;k 00 ;d 0 as given in (20); (iii) n is replaced by k00 ; and (iv) y 3 =S 3 is replaced by yb m;k 00 asgivenin(22). The primary drawback of the skewness-adjusted CI (34) is that except for the first spacer (which is potentially contaminated by transient effects), the other spacers contain useful information about the steadystate mean X that is ignored; and making efficient use of this information can yield more precise point and CI estimators of X. This consideration naturally leads to consideration of N-Skart s correlation adjustment Correlation adjustment to the NBM Student s t-statistic Correlation adjustment based on time series models of batch means Our correlation adjustment to the NBM Student s t-statistic is based on Theorems 1 3 and Lemma 1 below, coupled with the observation that in practice, many stationary Gaussian (normal) time series can be adequately modeled by a mixed autoregressive moving average (ARMA) process of order.p; q/, provided the autoregressive order p and the moving-average order q are chosen properly; see Box et al. (2008) and Priestley (1981, pp ). If the original (unbatched) process fx i g is a stationary and invertible ARMA.p 0 ;q 0 / process with 0 p 0 ;q 0 < 1, thentiao (1972) shows that as the batch size m! 1; the batch means fy j.m/g are asymptotically uncorrelated. The starting point for our analysis is Theorem 1 below, which extends the cited result of Tiao (1972) under a weaker set of hypotheses. Complete proofs of all results cited in this section are detailed in the Online Supplement to this article. Theorem 1. If fx i g is strictly stationary with mean X and variance parameter X, then for the nonoverlapping batch means fy j.m/g with batch size m, the lag-` correlation, `.m/ D CorrŒY j.m/; Y j C`.m/ for skart-iiet28.tex 14 June 20, :32

16 ` D 0; 1; 2;:::, satisfies lim `.m/ D 0 for ` D 1; 2;::: : (35) Moreover if the process fx i g is -mixing with mixing coefficients f i g satisfying P 1 id0 1=2 i < 1,then j `.m/jdj`.m/j2 1=2 0 for ` D 2;3;:::; and for a fixed batch count k, the standardized batch means as m!1:.` 1/mC1! n m=x 1=2 Yj.m/ X W j D 1;:::;k o are asymptotically i.i.d. standard normal variables m=x 1=2 Y1.m/ X ;:::;Y k.m/ X where 0 k denotes the k 1 null vector and I k denotes the k k identity matrix. D! N k.0 k ; I k /; (36) Sketch of Proof. Equation (35) is established by induction on ` for ` D 1;2;::: : From the well-known relation Var xy.m;`c1/ D VarŒY.m/ 1 C 2 ` C 1 `X ud1 1 u u.m/ ` C 1 for ` D 1;2;::: ; (37) it follows that for ` D 1 we have VarŒ xy.m;2/ VarŒY.m/ =2 1 D 1.m/ : (38) Because X D lim `mvarœ xy.m;`/ for ` D 1;2;:::, from Equation (38) it follows that lim 1.m/ D 0. For` D 2;3;:::, Equation (37) also yields the relation VarŒ xy.m;`c 1/ VarŒY.m/ =.` C 1/ VarŒ xy.m;`/ VarŒY.m/ =` D `.` C 1/ X` 1 ud1 3 u u.m/ 5 2 ` C 1 `.m/ I (39) and by induction on `, we can derive (35) ifweletm!1in Equation (39). The relation j`.m/j 2 1=2.` 1/mC1! 0 for ` D 2;3;::: follows from Equation (20.23) of Billingsley (1968), and Equation (36) is the main conclusion of Theorem 1 of Steiger and Wilson (2001). Equation (35) will play a key role in the rest of the development, and it is similar to Equation (2.8) of Tiao (1972). However, the only assumption required to obtain (35) isthatfx i g is stationary with a finite variance parameter. By contrast, the derivation of Equation (2.8) of Tiao (1972) is arguably more complicated and requires the stronger assumption that fx i g is a stationary and invertible ARMA(p 0 ;q 0 ) process for 0 p 0 ;q 0 < 1. All the foregoing considerations form the basis for our fundamental assumption that the underlying process fx i g is such that with a sufficiently large batch size m, the stochastic behavior of the associated batch means fy j.m/g can be accurately approximated by an ARMA.p; q/ process for finite nonnegative values of p and q. Specifically, we assume that for any batch size m m 0 (where m 0 is a base batch skart-iiet28.tex 15 June 20, :32

17 size taken sufficiently large), the batch means fy j.m/ W j D 1;2;:::g can be adequately modeled by an ARMA.p; q/ process Y j.m/ D X C px '`.m/ ŒY j `.m/ X C " j.m/ `D1 qx `.m/ " j `.m/ ; (40) where: p and q depend on m 0 ;wehave0 p; q < 1; the random shocks f" j.m/ W j D 1;2;:::g driving the process are randomly sampled from N 0; ".m/ 2 ; and the autoregressive parameters f'`.m/ W ` D 1;:::;pg and moving-average parameters f`.m/ W ` D 1;:::;qg in the ARMA model (40) satisfy the following conditions Stationarity: The roots of the degree-p autoregressive polynomial, ( 1; if p D 0; ˆp;m.z/ 1 P p '`.m/z` ; if p 1; `D1 n must lie outside the unit circle in the complex plane C z D x C y p o 1 W x;y 2 R. Invertibility: The roots of the degree-q moving-average polynomial, ( 1; if q D 0; q;m.z/ 1 P q `.m/z` ; if q 1; `D1 must lie outside the unit circle in C. Unicity: The polynomials ˆp;m.z/ and q;m.z/ have no common roots and no multiple roots. Moreover any two roots of ˆp;m.z/ or q;m.z/ that are not complex conjugates of each other must have distinct absolute values. Because both polynomials ˆp;m.z/ and q;m.z/ have real coefficients, their roots are either real numbers or conjugate pairs of complex numbers. In the unicity condition, the requirement that each polynomial has no multiple roots implies that its real roots must all have distinct values; moreover, the unicity condition stipulates that its nonconjugate roots must have distinct absolute values. The unicity assumption does not appear to be restrictive in practice, but it is necessary for the analysis that follows. See also Anderson (1971, p. 236) andbox et al. (2008, p. 58). If p>0, then for m m 0 we let fr i.m/ 2 C W i D 1;:::;pg denote the roots of ˆp;m.z/ so that we can write py ˆp;m.z/ D Œ1 ı i.m/z for all z 2 C ; (41) id1 where ı i.m/ D Œr i.m/ 1 2 C and jı i.m/j <1for i D 1;:::;p by the stationarity condition. Similarly if q>0, then for m m 0 we let fu j.m/ 2 C W j D 1;:::;qg denote the roots of q;m.z/ so that we can write qy q;m.z/ D Œ1! j.m/z for all z 2 C ; (42) j D1 skart-iiet28.tex 16 June 20, :32 `D1

18 where! j.m/ D Œu j.m/ 1 2 C and j! j.m/j <1for j D 1;:::;q by the invertibility condition. In view of Theorem 1, we make the following continuity assumption about the autoregressive and moving-average coefficients in (40): ) lim ' i.m/ D 0 for i D 1;:::;p (provided p>0) ; (43) lim j.m/ D 0 for j D 1;:::;q (provided q>0) : The continuity assumption is critical to the development of Theorem 3 below, which is the basis for the correlation adjustment used in Skart. Although we cannot prove that (40) and(43) hold for every stationary stochastic process with a finite variance parameter, we can establish (40) and(43) in a large class of stationary stochastic processes that at least make (40) and(43) plausible assumptions about any batch-means process fy j.m/g obtained by aggregating (batching) a stationary process, provided the batch size m is sufficiently large. Theorem 2. If the original (unbatched) process fx i g is a stationary and invertible ARMA(p 0 ;q 0 ) process, Xp 0 Xq 0 X i D X C '`.X i ` X / C " i ` " i ` ; (44) `D1 where 0 p 0 ;q 0 < 1, then the batch-means process fy j.m/ W j D 1;2;:::g is a stationary and invertible ARMA(p; q) process (40) with ( ) p 0 ; if q 0 <p 0 C 1; p D p 0 and q D for m m 0 Djp 0 q 0 jc1: (45) p 0 C 1; if q 0 p 0 C 1; Moreover, the autoregressive and moving-average coefficients in the ARMA(p; q) model describing the batch-means process fy j.m/g satisfy the continuity condition (43). Sketch of Proof. To avoid trivial cases, we assume that p 0 1 and q 0 1. It is straightforward to adapt the following argument to handle situations in which p 0 D 0 or q 0 D 0 (or both). The result (45) is established by Brewer (1973, p. 145). Corresponding to the ARMA(p 0 ;q 0 ) representation (44) ofthe original (unbatched) process fx i g,weletfı W D 1;:::;p 0 g denote the inverse roots of the associated autoregressive polynomial, ( 1; if p 0 D 0; ˆp0.z/ 1 P p 0 `D1 '`z`; if p 0 1 I and we let f! W D 1;:::;q 0 g denote the inverse roots of the associated moving-average polynomial, `D1 Because (44) is stationary and invertible, we have ( 1; if q0 D 0; q0.z/ 1 P q 0 `D1 `z`; if q 0 1: jı j <1 for D 1;:::;p 0 (provided p 0 >0) j! j <1 for D 1;:::;q 0 (provided q 0 >0) ) : (46) skart-iiet28.tex 17 June 20, :32

19 If the original (unbatched) process has the ARMA(p 0 ;q 0 ) representation (44), then from the second (unnumbered) equation on p. 467 of Silvestrini and Veredas (2008) we see that in the ARMA(p; q) representation of the batch-means process fy j.m/g with batch size m m 0 Djp 0 q 0 jc1 and autoregressive and moving-average orders p and q givenby(45), the `th autoregressive parameter '`.m/ equals. 1/`C1 times the sum of products of the mth powers of individual elements from the set fı W D 1;:::;pg taken ` at a time for ` D 1;:::;p. The desired continuity condition (43) for the autoregressive coefficients f'`.m/ W ` D 1;:::;pg then follows immediately from (46). To prove that the moving-average coefficients f`.m/ W ` D 1;:::;qg also satisfy the continuity condition (43), a less straightforward approach is required. Computing batch means from the moving-average component of (44), we obtain the auxiliary process U j.m/ D 1 m jmx id.j 1/mC1 " i q 0 X `D1 ` " i ` ; for j D 1;2;::: : (47) From the analysis concerning the second (unnumbered) equation on p. 467 of Silvestrini and Veredas (2008), we see that (47) defines an invertible MA.q/ process whose autocorrelation function, z`.m/ D CorrŒU j.m/; U j C`.m/ for ` D 0; 1; 2;::: ;satisfies z`.m/ 6D 0 if ` D q;and z`.m/ D 0 if j`j >q. Our approach is based on the analysis given in of Anderson (1971), which we will adapt to show that given the marginal variance VarŒU j.m/ and the associated correlations z 1.m/,..., z q.m/, the invertible MA(q) representation of the auxiliary process fu j.m/g defined by (47) has moving-average coefficients f`.m/ W ` D 1;:::;qg that converge to zero as m! 1. We consider the roots of the degree-(2q) polynomial C m.z/ D 2q X `D0 z` q.m/z` : As shown in of Anderson (1971), the roots of C m.z/ D 0 can be divided into two sets fz!`.m/ W ` D 1;:::;qg and fz! qc`.m/ W ` D 1;:::;qg such that ( ) if jz!`.m/j <1; then z! qc`.m/ D 1=z!`.m/ For ` D 1;:::;q; we have jz!`.m/j 1; and if jz!`.m/j D1; then z! qc`.m/ Dz!`.m/ : Moreover, from the analysis in of Anderson (1971), we see that moving-average polynomial corresponding to the MA(q) representation of the process fu j.m/ W j D 1;2;:::g has the form qx qy q;m.z/ 1 `.m/z` D Œ1 z!`.m/z (48) `D1 `D1 because ( C m.z/ D z q.m/ ; if z D 0; z q q;m.z/ q;m.1=z/ ; if z 6D 0 I and from (48), we see immediately that in general `.m/ equals. 1/`C1 times the sum of products of individual elements from the set fz!.m/ W D 1;:::;qg taken ` at a time for ` D 1;:::;q. The desired skart-iiet28.tex 18 June 20, :32

20 continuity condition (43) for the moving-average coefficients f`.m/ W ` D 1;:::;qg will follow immediately if we can show that lim jz!.m/j D0 for D 1;:::;q: To establish the latter result, we will prove that the quantity b.m/ minfjz! qc.m/j W D 1;:::;qg satisfies lim b.m/ DC1: Suppose on the contrary that lim inf b.m/ D b < 1 : (49) Choose >0arbitrarily. From (49) it follows immediately that we can find at least one integer 2 f1;:::;qg together with a strictly increasing subsequence fm./ W D 1;2;:::g of batch sizes such that lim!1 m./ DC1and 1 jz! qc Œm./ j b C D b for D 1;2;::: : (50) For real numbers R 1 and R 2 with 0 R 1 < R 2,letA.R 1 ;R 2 / fz 2 C W R 1 jzj R 2 g denote the annulus centered at the origin with inner radius R 1 and outer radius R 2. From (50) wehave fz! qc Œm./ W D 1;2;:::gA.1; b /. Because A.1; b / is closed and bounded (and hence compact), the Bolzano-Weierstrass Theorem (Knopp, 1952) implies that fz! qc Œm./ W D 1;2;:::g has a limit point 2 A.1; b /. This means that there exists a subsubsequence fz! qc Œm. h / W h D 1;2;:::g of the subsequence fz! qc Œm./ W D 1;2;:::g such that lim h!1 z! qc Œm. h / D 2 A.1; b /: We have C m.h /fz! qc Œm. h / g D0 for h D 1;2;::: ; (51) since for each m m 0, recall that z! qc.m/ is a root of C m.z/ D 0. On the other hand, we must have lim C m.h / z!qc Œm. h / D lim h!1 2qX h!1 `D0 X2q n D `D0 lim h!1 z` q Œm. h / z! qc Œm. h / ` o z` q Œm. h / lim h!1 z!qc Œm. h / ` D q (52) because z 0.m/ 1 and lim z`.m/ D 0 for ` D 1; 2;:::; q by Equation (35) of Theorem 1. Because we must have jj 1 by the definition of A.1; b /, Equation (52) contradicts (51); and thus the assumption (49)mustbefalseforb < 1. It follows that 1Dlim inf b.m/ lim sup b.m/ so that we have lim b.m/ DC1Iand thus the desired conclusion of Theorem 2 follows. The following technical result is needed in the rest of this subsection, and it follows from the assumption (40) that the batch-means process is ARMA for a sufficiently large batch size together with the continuity assumption (43). The proof of Lemma 1 uses techniques similar to those of Theorem 2; the detailed proof is provided in the Online Supplement to this article. Lemma 1. If fy j.m/g is a stationary and invertible ARMA(p; q) process (40)form m 0 and the continuity condition (43) holds, then ) lim jı i.m/j D0 for i D 1;:::;p (provided p>0) ; (53) lim j! j.m/j D0 for j D 1;:::;q (provided q>0) : skart-iiet28.tex 19 June 20, :32

21 To complete the analysis of the batch-means process fy j.m/ W j D 1;2;:::g for sufficiently large values of the batch size m, we impose the following regularity condition on the way in which the inverse autoregressive roots fı i.m/ W i D 1;:::;pg and the inverse moving-average roots f! j.m/ W j D 1;:::;qg tend to zero in absolute value as m!1: lim sup jı i.m/j j P p < 1 for i D 1;:::;p (provided p>0) (54) ı`.m/j `D1 and j! j.m/j lim sup j P q < 1 for j D 1;:::;q (provided q>0) : (55)!`.m/j `D1 Equation (53) in Lemma 1 ensures that both the numerators and denominators in (54) and(55) tend to zero as m! 1. The condition (54) merely requires that the absolute value of the sum of inverse autoregressive roots should tend to zero as m!1at a rate which does not differ too much from the rates at which the absolute values of each of the individual inverse autoregressive roots tend to zero. Condition (55) imposes a similar requirement on the rates at which the absolute values of the inverse moving-average roots tend to zero as m!1. In particular, inverse autoregressive roots that decline in absolute value according to an inverse-power law in the batch size m or exponentially in the batch size as m!1and that satisfy the unicity condition can be easily shown to satisfy (54); and a similar statement provides common situations in which the inverse moving-average roots are easily shown to satisfy (55). In the following statement of Theorem 3, we use the little-oh notation g.m/ D oœh.m/ to mean that lim n!1 g.m/=h.m/ D 0. Theorem 3. If for all m m 0 the batch-means process fy j.m/ W j D 1;2;:::g is an ARMA.p; q/ process (40) that satisfies the stationarity, invertibility, and unicity conditions as well as the continuity condition (43) and the regularity conditions (54) and (55), then as m!1wehave '.m/ D oœ' 1.m/ for D 2;:::;p (provided p>1) ; (56).m/ D oœ 1.m/ for D 2;:::;q (provided q>1) ; (57) so that Var xy.m;k 0 / VarŒY.m/ k 0 VarŒY.m/ k 0 1 ' 1 2.m/ 1 1.m/ 2 1 2' 1.m/ 1.m/ C 1 2.m/ 1 ' 1.m/ as m!1: (58) 1 C '1.m/ 1 ' 1.m/ Sketch of Proof. To avoid trivial cases, throughout the rest of this proof we assume that p 2 and q 2. It is straightforward to adapt the following argument to handle situations in which p 1 or q 1 (or both). Corresponding to the ARMA(p; q) model of the batch means fy j.m/g for m m 0, we see that in general for the autoregressive polynomial (41), its `th coefficient '`.m/ equals. 1/`C1 times the sum of products of individual elements from the set fı.m/ W D 1;:::;pg taken ` at a time for ` D 1;:::;p. Similarly, in general for the moving-average polynomial (42), the `th coefficient `.m/ equals. 1/`C1 times the sum of products of individual elements from the set f!.m/ W D 1;:::;qg taken ` at a time for ` D 1;:::;q. skart-iiet28.tex 20 June 20, :32