Automatic batching in simulation output analysis

Antonio Manuel de Carvalho Pedrosa, Purdue University

Abstract

We develop methodology and propose an algorithm to estimate the variance of the sample mean of stationary stochastic processes. The algorithm, denoted by 1-2-1 OBM, is (i) automatic, in that it does not require the user to provide any parameter, (ii) robust, in that it can be applied to any stationary stochastic process and is not sensitive to reasonable violations of assumptions, (iii) computationally efficient, in that the computation time is proportional to the sample size, and (iv) statistically efficient, in that the mean squared error performance is good. We also provide a FORTRAN implementation. Our approach minimizes the mean squared error of estimators of the variance of the sample mean parameterized by batch size by estimating the optimal batch size from a given data sample. The optimal batch size is a simple function of the correlation structure of the data: the sum of autocorrelations, $\gamma\sb0$, and the weighted sum of autocorrelations, $\gamma\sb1$. We develop estimators $\\gamma\sb0$ and $\\gamma\sb1$ of $\gamma\sb0$ and $\gamma\sb1$. Both $\\gamma\sb0$ and $\\gamma\sb1$ are obtained from overlapping-batch-means (OBM) estimators, but analogous development can be considered for other estimators parameterized by batch size. We develop theoretical and empirical guidelines to estimate the optimal batch size from $\\gamma\sb0$ and $\\gamma\sb1$. We prove that our optimal batch size estimator converges in probability to the optimal batch size. We also develop asymptotically unbiased methodology to estimate the variance of the sample mean based on classical linear regression of OBM estimators. This approach motivates our derivation of asymptotic results regarding bias, covariance, and correlation of OBM estimators. We derive analogous results for Bartlett estimators, and we generalize these results for non-zero frequencies, i.e., for Bartlett estimators of the spectral density. We prove that OBM estimators and Bartlett estimators of the variance of the sample mean are asymptotically equivalent. In addition, we develop DPSS, a four-parameter family of stochastic processes with good analytical, computational, and statistical properties for evaluating simulation methods. The four parameters are related to the mean, variance, lag-one autocorrelation, sum of autocorrelations, and weighted sum of autocorrelations through simple closed-form equations. The stationary marginal distribution is discrete uniform. Variates are easily and efficiently generated. The autocorrelation structure has a damped oscillating behavior allowing for a wide range of dependency characteristics.

Degree

Ph.D.

Advisors

Schmeiser, Purdue University.

Subject Area

Industrial engineering|Operations research|Statistics

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