On improved estimation for importance sampling

The standard estimator used in conjunction with importance sam- pling in Monte Carlo integration is unbiased but inefficient. An alternative estimator is discussed, based on the idea of a difference estimator, which is asymptotically optimal. The improved estimator uses the importance weight as a control variate, as previously studied by Hesterberg (Ph.D. Disserta- tion, Stanford University (1988); Technometrics 37 (1995) 185-194; Statis- tics and Computing 6 (1996) 147-157); it is routinely available and can de- liver substantial additional variance reduction. Finite-sample performance is illustrated in a sequential testing example. Connections are made with meth- ods from the survey-sampling literature.