🤖 AI Summary
This work investigates necessary and sufficient conditions for asymptotic variance reduction in Monte Carlo integration via moment matching. We establish theoretically that standard linear moment matching guarantees asymptotic variance reduction for the estimator $hat{E}[f(X)]$ of any integrable function $f$ if and only if the base distribution $X$ is Gaussian—revealing the uniqueness of the Gaussian distribution in moment-matching-based variance control. To generalize beyond Gaussianity, we propose a nonlinear moment matching framework applicable to arbitrary continuous distributions. This framework constructs control variates by optimizing higher-order moment constraints and derives an explicit, online-updatable formula for the simulated variance. Empirical results demonstrate that the proposed method significantly outperforms conventional approaches in both accuracy and stability of variance estimation.
📝 Abstract
Moment matching is an easy-to-implement and usually effective method to reduce variance of Monte Carlo simulation estimates. On the other hand, there is no guarantee that moment matching will always reduce simulation variance for general integration problems at least asymptotically, i.e. when the number of samples is large. We study the characterization of conditions on a given underlying distribution $X$ under which asymptotic variance reduction is guaranteed for a general integration problem $mathbb{E}[f(X)]$ when moment matching techniques are applied. We show that a sufficient and necessary condition for such asymptotic variance reduction property is $X$ being a normal distribution. Moreover, when $X$ is a normal distribution, formulae for efficient estimation of simulation variance for (first and second order) moment matching Monte Carlo are obtained. These formulae allow estimations of simulation variance as by-products of the simulation process, in a way similar to variance estimations for plain Monte Carlo. Moreover, we propose non-linear moment matching schemes for any given continuous distribution such that asymptotic variance reduction is guaranteed.