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The inefficiency of using an unbiased estimator in a Monte Carlo procedurecan be quantified using an inefficiency constant, equal to the product of thevariance of the estimator and its mean computational cost. We develop methodsfor obtaining the parameters of the importance sampling (IS) change of measurevia single- and multi-stage minimization of well-known estimators ofcross-entropy and the mean square of the IS estimator, as well as of newestimators of such a mean square and inefficiency constant. We prove theconvergence and asymptotic properties of the minimization results in ourmethods. We show that if a zero-variance IS parameter exists, then, underappropriate assumptions, minimization results of the new estimators converge tosuch a parameter at a faster rate than such results of the well-knownestimators, and a positive definite asymptotic covariance matrix of theminimization results of the cross-entropy estimators is four times such amatrix for the well-known mean square estimators. We introduce criteria forcomparing the asymptotic efficiency of stochastic optimization methods,applicable to the minimization methods of estimators considered in this work.In our numerical experiments for computing expectations of functionals of anEuler scheme, the minimization of the new estimators led to the lowestinefficiency constants and variances of the IS estimators, followed by theminimization of the well-known mean square estimators, and the cross-entropyones.

Adaptive importance sampling via minimization of estimators of cross-entropy, mean square, and inefficiency constant