Halim Damerdji

Mean-Square Consistency of the Variance Estimator in Steady-State Simulation Output Analysis

In steady-state simulation output analysis, mean-square consistency of the process-variance estimator is important for a number of reasons. One way to construct an asymptotically valid confidence interval around a sample mean is via construction of a consistent estimator of the process variance and a central limit theorem. Also, if an estimator is consistent in the mean-square sense, a mean-square error analysis is theoretically justified. Finally, batch-size selection is an open research problem in steady-state output analysis, and a mean-square error analysis approach has been proposed in the literature; to be valid, the process-variance estimators constructed must be consistent in the mean-square sense. In this paper, we prove mean-square consistency of the process-variance estimator for the methods of batch means, overlapping batch means, standardized time series area, and spaced batch means, by rigorously computing the rate of decay of the variance of the process-variance estimators. Asymptotic results for third and higher centered moments of the batch means and area variance estimators are also given, along with central limit theorems.

Two-stage procedures for multiple comparisons with a control in steady-state simulations

Suppose that we have k different stochastic systems, where /spl mu/i denotes the steady-state mean of system i. We assume that the system labeled k is a control and want to compare the performance of the other sys tems, labeled 1,2,...,k - 1, relative to this control. This problem is known in the statistical literature as multiple comparisons with a control (MCC). Independent steady-state simulations will be performed to compare the systems to the control. Two-stage procedures, based on the method of batch means, are presented to construct simultaneous lower one sided confidence intervals for/spl mu/i - /spl mu/k (i = 1, 2, . . ., k), each having prespecified (absolute or relative) half width 6. Under the assumption that the stochastic processes representing the evolution of the systems satisfy a functional central limit theorem, it can be shown that asymptotically (as /spl delta/ /spl rarr/ 0 with the size of the batches proportional to 1//spl delta//sup 2/), the joint probability that the confidence intervals simultaneously contain the /spl mu/i - /spl mu/k (i = 1, 2,..., k - 1) is at least 1 - /spl alpha/, where /spl alpha/ is prespecified by the user.

Selecting the best system in transient simulations with variances known

Selection of the best system among k different systems is investigated. This selection is based upon the results of finite-horizon simulations. Since the distribution of the output of a transient simulation is typically unknown, it follows that this problem is that of selection of the best population (best according to some measure) among k different populations, where observations within each population are independent, and identically distributed according to some general (unknown) distribution. In this work in progress, it is assumed that the population variances are known. A natural single-stage sampling procedure is presented. Under Bechbofers indifference zone approach, this procedure is asymptotically valid.

Estimating and simulating Poisson processes with trends or asymmetric cyclic effects

We present a heuristic that provides a nonparametric estimate of the mean-value function of a nonhomogeneous Poisson process having a long-term trend or some cyclic effect(s) that may lack sinusoidal symmetry over the corresponding cycle(s). This heuristic is a multiresolution-based method that allows one to estimate the overall long-term trend of the process at the lowest resolution and then add the details of the process associated with progressively smaller periodic components at progressively higher resolutions. In addition, we present an algorithm for generating realizations of nonhomogeneous Poisson processes with the estimated mean-value function in simulation experiments.

Consistency of several variants of the standardized time series area variance estimator

In statistical analysis of stationary time series or in steady-state simulation output analysis, it is desired to find consistent estimates ofthe process variance parameter. Here, we consider variants ofthe area estimator of standardized time series, namely, the weighted area and the Cramer-von Mises area estimators, and provide their consistency, in the strong sense and mean-square sense. A sharp bound for the (asymptotic) variance of these estimators is obtained. We also present a central limit theorem for the weighted area estimator : this gives a rate of convergence of this estimator, as well as a confidence interval for the variance parameter.

Computational efficiency evaluation in output analysis

A central quantity in steady-state simulation is the time-average variance constant. Estimates of this quantity are needed (for example) for constructing confidence intervals, and several estimators have been proposed, including nonoverlapping and overlapping batch means methods, spectral methods, and the regenerative method. The asymptotic statistical properties of these estimators have been investigated but the computational complexity involved in computing them has received very little attention. We assume a fixed simulation run-length, as opposed to sequential methods in which the run-length is determined dynamically. In order to consistently estimate the time-average variance constant, all of the estimators require an amount of computation that is linear in the time-horizon simulated, with the exception of spectral methods which require a superlinear amount of computation.

On the batch means and area variance estimators

In steady-state simulation output analysis, construction of a consistent estimator of the the variance parameter of the process may be desirable in a number of instances. For example, if the process obeys a central limit theorem and an estimator of the variance is available, one may, then, construct an asymptotically valid confidence interval for the process-mean parameter. Centered moments (e.g. bias, variance, skewness, etc.) of an estimator are familiar measures of goodness of that estimator. Also, a central limit theorem involving the estimator provides its asymptotic rate of convergence. We consider here the batch means and the standardized time series area variance estimators, in their nonclassical setting, and provide asymptotic expressions for their centered moments as well as central limit theorems. As a by-product, consistency in the mean-square sense of these estimators is obtained. Our assumption on the process does not include stationarity nor covariance stationarity (although we are in the steady-state context).

Performance analysis of future event sets

The linked list and indexed list future event sets are investigated. The interaction hold model and the Jackson network model are the underlying stochastic models considered. For the interaction hold model and for the (doubly) linked list, we find, for example, the mean number of key comparisons performed in order to find a records insertion point into the list; this is useful when deciding whether to scan from the head or the tail of the list. The distribution of the relative position of the to-be-inserted record is also obtained; for indexed lists, this is helpful when deciding the number of sublists and position(s) of the middle pointer(s). The Jackson network model has a realistic event logic, but events are restricted to be exponentially distributed. Because the stationary probabilities can be computed for this model, it is then possible to evaluate and compare the (steady-state) performance of certain future event sets (e.g. linked lists scanned from the head or the tail).