Variance reduction for quantile estimates in simulations via nonlinear controls

Abstract

Linear controls are a well known simple technique for achieving variance reduction in computer simulation. Unfortunately the effectiveness of a linear control depends upon the correlation between the statistic of interest and the control, which is often low. Since statistics often have a nonlinear relationship with the potential control variables, nonlinear controls offer a means for improvement over linear controls. This paper focuses on the use of nonlinear controls for reducing the variance of quantile estimates in simulation. It is shown that one can substantially reduce the analytic effort required to develop a nonlinear control from a quantile estimator by using a strictly monotone transformation to create the nonlinear control. It is also shown that as one increases the sample size for the quantile estimator, the asymptotic multivariate normal distribution of the quantile of interest and the control reduces the effectiveness of the nonlinear control to that of the linear control. However, the data has to be sectioned to obtained an estimate of the variance of the controlled quantile estimate. Graphical methods are suggested for selecting the section size that maximizes the effectiveness of the nonlinear control. Keyword: Variance reduction, Quantiles; Nonlinear controls; Transformation; ACE; Least-squares regression; Jackknifing. (kr)