Post-processing accept-reject samples: Recycling and rescaling

This article proposes alternative methods for constructing estimators from accept-reject samples by incorporating the variables rejected by the algorithm, The resulting estimators are quick to compute, and turn out to be variations of importance sampling estimators, although their derivations are quite different. We show that these estimators are superior asymptotically to the classical accept-reject estimator, which ignores the rejected variables. In addition, we consider the issue of rescaling of estimators, a topic that has implications beyond accept-reject and importance sampling. We show how rescaling can improve an estimator and illustrate the domination of the standard importance sampling techniques in different setups.