  The inefficiency of using an unbiased estimator in a Monte Carlo procedure
can be quantified using an inefficiency constant, equal to the product of the
variance of the estimator and its mean computational cost. We develop methods
for obtaining the parameters of the importance sampling (IS) change of measure
via single- and multi-stage minimization of well-known estimators of
cross-entropy and the mean square of the IS estimator, as well as of new
estimators of such a mean square and inefficiency constant. We prove the
convergence and asymptotic properties of the minimization results in our
methods. We show that if a zero-variance IS parameter exists, then, under
appropriate assumptions, minimization results of the new estimators converge to
such a parameter at a faster rate than such results of the well-known
estimators, and a positive definite asymptotic covariance matrix of the
minimization results of the cross-entropy estimators is four times such a
matrix for the well-known mean square estimators. We introduce criteria for
comparing 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 an
Euler scheme, the minimization of the new estimators led to the lowest
inefficiency constants and variances of the IS estimators, followed by the
minimization of the well-known mean square estimators, and the cross-entropy
ones.
