Large-sample normality of the batch-means variance estimator

Abstract

Consider a stationary stochastic process, X 1, X 2,…, arising from a steady-state simulation. An important problem is that of estimating the expected value μ of the process. The usual estimator for μ is the sample mean based on n observations, X ̄ n , and a measure of the precision of X ̄ n is the variance parameter, σ 2= lim n→∞ n Var[ X ̄ n] . This paper studies asymptotic properties of the batch-means estimator V ̂ B (b,m) for σ 2 as both the batch size m and number of batches b become large. In particular, we give conditions for V ̂ B (b,m) to converge to normality as m and b increase. Empirical examples illustrate our findings.