Mathematics of Operations Research February 1990 [antikvár]

MATHEMATICS OF OPERATIONS RESEARCH Vol. 15. No. I. February 1990 Primed hi U.S.A. SIMULATION OUTPUT ANALYSIS USING STANDARDIZED TIME SERIES*^^ PETER W. GLYNN and DONALD L. IGLEHART Stanford University The method of standardized time series (STS) was proposed by Schruben as an approach for constructing asymptotic confidence intervals for the steady-state mean from a single simulation run. The STS method "cancels out" the variance constant while other methods attempt to consistently estimate the variance constant. Our goal in this paper is to generalize the STS method and to study some of its basic properties. Starting from a functional central limit theorem (FCLT) for the sample mean of the simulated process, a class of mappings of C[0,1] to U is identified, each of which leads to a STS confidence interval. One of these mappings leads to the batch means method. A lower bound is obtained for the expected length of the asymptotic (as the run size becomes large) STS confidence intervals. This lower bound is not attained, but can be approached arbitrarily closely, by STS confidence intervals. Methods that consistently estimate the variance constant do realize this lower bound. The variance of the length of a STS confidence interval is of larger order (in the run length) than is that for the regenerative method. 1. Introduction. A principal problem in the simulation literature is to construct asymptotic (as the run length becomes large) confidence intervals for steady-state parameters of the simulation output process from a single simulation run. There are two basic approaches to this problem. The first is to consistently estimate the variance constant in the relevant central limit theorem. This is the approach used in the regenerative, spectral, and autoregressive methods. The second approach, proposed by Schruben (1983), is based on standardized time series (STS), and essentially "cancels out" the variance constant in a manner reminiscent of the i-statistic. For other work on STS see Chen and Sargent (1985), Goldsman and Schruben (1984), Glynn and Iglehart (1985), and Nozari (1986). Our goal in this paper is to generalize the method of STS and to study some of its basic properties. The starting point for the STS method is the existence of a functional central limit theorem (FCLT) for the sample mean of the simulated process. These FCLT's exist for stationary (and some nonstationary) (f>-mixing processes, strictly stationary strongly mixing processes, associated strictly stationary processes, and regenerative processes. We identify a class of mappings from C[0,1] to IR, each of which leads to a STS confidence interval. One of these mappings yields the batch means method. We study the asymptotic length of these STS confidence intervals and develop a lower bound for the expected length; see (4.16). This lower bound is attained by the methods which consistently estimate the variance constant. While this lower •Received June 26, 1985; revised August 22, 1988. AMS 1980 subject classification, primary; 65C05. Secondary; 62M10, lAOR 1973 subject classification. Main; Simulation, Cross references; Statistics; inference, OR/MS Index 1978 subject classiftcation. Primary; 767 Simulation/Statistical analysis. Secondary; 805 Statistics/time series. Key words. Batch means, confidence intervals, functional central limit theorem, weak convergence of probability measures, simulation output analysis, standardized time series, steady-state simulation, *This work was supported by Army Research Office Contract DAAG29-84-K-0030 and National Science Foundation Grants MCS-82-3483, DCR-85-09668, and ECS-84-04809,