Control Variates for Probability and Quantile Estimation

Abstract

In stochastic systems, quantiles indicate the level of system performance that can be delivered with a specified probability, while probabilities indicate the likelihood that a specified level of system performance can be achieved. We present new estimators for use in simulation experiments designed to estimate such quantiles or probabilities of system performance. All of the estimators exploit control variates to increase their precision, which is especially important when extreme quantiles (in the tails of the distribution of system performance) or extreme probabilities (near zero or one) are of interest. Control variates are auxiliary random variables with known properties—in this case, known quantiles—and a strong stochastic association with the performance measure of interest. Since transforming a control variate can increase its effectiveness, we propose both continuous and discrete approximations to the optimal (variance-minimizing) transformation for estimating probabilities, and then invert the probability estimators to obtain corresponding quantile estimators. We also propose a direct control-variate quantile estimator that is not based on inverting a probability estimator. An empirical study using queueing, inventory and project-planning examples shows that substantial reductions in mean squared error can be obtained when estimating the 0.9, 0.95, and 0.99 quantiles.