Interlaced Estimators of the Marginal Variance in Steady-state Simulation

Motivated by steady-state simulation experiments, we consider the problem of estimating the marginal variance of a stationary time series. The usual estimator, the sample variance is biased for autocorrelated data. To reduce bias, other authors have suggested interlaced estimators. These estimators which like the sample variance are sums of squares are a generalization of the sample variance and a special case of quadratic forms. We show that, despite their smaller bias interlaced estimators have larger mean squared error than the sample variance. In addition we show that general quadratic forms provide little statistical advantage over the computationally less expensive sums-of-squares estimators. We conclude that the sample variance should he used in practice. ) 1 ( 1 − n n 2 1. Introduction We consider estimating the marginal variance, σ=V(Xi), from stationary time series data {Xi: i=1,2,…,n}. Estimation of the marginal variance is important for a number of reasons. In the context of quality control, the variance is necessary to set the tolerance limits. In experimental design studies it is used to test the difference or indifference of response parameter(s) at different design points. In simulation Studies it is a performance measure used to compare system designs. The usual estimator of the population variance is the sample variance. S = ∑ j=1, (Xj Χ ) / (n 1), where is the sample mean. The expected value of the sample variance as a function of the variance of the sample mean is (from [7], for example) E (S)= 1 − n n (σ V ( ( ) Χ ). Since V ( ) Χ is equal to σ/n for identically and independently distributed (iid) data and since it goes to zero for autocorrelated data, the sample variance is unbiased for iid data and asymptotically unbiased for autocorrelated data. The sample variance is a quadratic-form estimator; that is, it can be written as S = ∑ ∑ = = n