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Referring to Monte Carlo integration, different algorithms yield different estimators for the integral. A relevant issue is which algorithm is more efficient. The efficiency issue can be examined from the statistical properties of the estimator from a given algorithm and its computational aspects. Rubinstein (1981) showed a practical measure of the efficiency of an algorithm by t x Var(©), with t being the computer time required to compute ©, which estimates ©. Algorithm 1 is more efficient than algorithm 2 if
If the computational times for the two algorithms are approximately equal, comparison of efficiency can be made by examining the relative magnitude of the variances. When the true variances are not known, which is generally the case, sample variances can be used. Without considering the computational time, it can be shown that the sample-mean algorithm using X ~ U(a, b) is more efficient than the hit-and-miss algorithm (see Problem 6.25).
