Efficient Computation of Overlapping Variance Estimators for Simulation

Abstract

F or a steady-state simulation output process, we formulate efficient algorithms to compute certain estimators of the process variance parameter (i.e., the sum of covariances at all lags), where the estimators are derived in principle from overlapping batches separately and then averaged over all such batches. The algorithms require order-of-sample-size work to evaluate overlapping versions of the area and Cramér-von Mises estimators arising in the method of standardized time series. Recently, Alexopoulos et al. showed that, compared with estimators based on nonoverlapping batches, the estimators based on overlapping batches achieve reduced variance while maintaining similar bias asymptotically as the batch size increases. We provide illustrative analytical and Monte Carlo results for M/M/1 queue waiting times and for a first-order autoregressive process. We also present evidence that the asymptotic distribution of each overlapping variance estimator can be closely approximated using an appropriately rescaled chi-squared random variable with matching mean and variance.

Experimental Evaluation of Integrated Path Estimators

“…The model and these calculations are described in Alexopoulos, Argon, Goldsman, Steiger, Tokol, and Wilson (2005).…”

SPSTS: A sequential procedure for estimating the steady-state mean using standardized time series

“…The idea behind this truncation is that once the signed areas pass the randomness test in steps (8) and (10). The accuracy of the approximation (10) was assessed experimentally by Alexopoulos et al (2007a).…”

“…• Although estimation of the variance parameter σ 2 is generally much more difficult than estimation of the steady-state mean µ (see, for example, Alexopoulos et al, 2007a andLada et al, 2007), SPSTS has the potential to deliver readily both a point estimator and a valid CI estimator of σ 2 . This property distinguishes SPSTS from its state-of-the-art NBM-based competitors.…”

Monitoring autocorrelated processes using a distribution-free tabular CUSUM chart with automated variance estimation

“…Other weight functions and even other STS estimators for Ω 2 are available for use; our selection here has been based on the comparatively good analytical and empirical performance of the overlapping area estimator defined by Equation (10) For use with DFTC-VE, we propose an automated batch-size determination algorithm that uses the same sequential procedure as in Lada and Wilson (2006) and Lada et al (2008); but instead of using nonoverlapping batch means as the basic data items to be tested for independence and normality, we use the "STS-weighted-area" statistics similar to those defined in Equation (9) Remark 4. The first part of Equation (11) (10) has approximately a scaled chi-squared distribution with at least 48 degrees of freedom (Alexopoulos et al, 2007a).…”

“…Fortunately, the simulation literature provides a number of variance-estimation techniques based on the following methods for analysis of steady-state simulation outputs: autoregressive representation (Fishman, 1971); nonoverlapping batch means (Fishman and Yarberry, 1997); overlapping batch means (Alexopoulos et al, 2007a); spectral analysis (Lada and Wilson, 2006); and standardized time series (STS) (Schruben, 1983). Although accurate and efficient estimation of the variance parameter is an important research problem by itself, in this article we are more interested in developing automated variance-estimation procedures that can be effectively incorporated into distribution-free SPC charts.…”