Optimal quadratic-form estimator of the variance of the sample mean

Abstract

A classical problem of stochastic simulation is how to estimate the variance of the sample mean of dependent but stationary outputs. Many variance estimators, such as the batch means estimators and spectral estimators, can be classified as quadratic-form estimators.Necessary and sufficient conditions on the quadratic-form coefficients such that the corresponding variance estimator has good performance have been proposed. But no one has utilized these conditions to pursue optimal quadratic-form coefficients to form an optimal variance estimator. In this paper, we seek an optimal (minimum variance unbiased) variance estimator by searching for the optimal quadratic-form coefficients.

Simulation output analysis using integrated paths II

“…One approach to constructing an estimator for σ 2 is to apply a quadratic function to the output vector Y n ; see for example Song et al [1997]. In general, the cost of computing such an estimator is proportional to the square of the simulation run length, n. An example of this approach is the batch means estimator, though in the case of batch means the complexity is linear in run length due to the special form of the quadratic function.…”

“…For k = 1, the integrated path estimator is the area estimator [Schruben 1983]. Song et al [1997] consider the problem of applying a quadratic function to the entire simulation output. Their work resulted in optimization problems similar to the ones we encounter in Section 3.…”