Efficient Scores, Variance Decompositions, and Monte Carlo Swindles

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Abstract
Monte Carlo “swindles” or variance reduction techniques exploit the experimenter's knowledge of the stochastic structure governing the simulated data to construct more precise estimates of unknown parameters. Alternatively, one can reduce the number of replications (and thus the cost) needed to gain a desired level of precision. This article reviews the common case of swindles based on variance decompositions for estimating efficiencies and variances of location and regression estimators. We then propose a new swindle based on Fisher's efficient score function that can be applied profitably to a much wider range of situations than can the Gaussian-over-independent swindles used in many studies of robust estimators. We compare these methods by performing simulations for the efficiencies of location estimates and by placing them in a simple geometric framework. As an illustration, the score function swindle is used to estimate the variances of Pitman estimates of location for samples from selected t-distributions. Finally, we sketch applications to scale estimation, exponential regression, statistical decision theory, and bootstrap computations.

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