Variance Reduction via Regression-Adjusted Monte Carlo for the Weitzman Overlap Coefficient

The Weitzman overlap coefficient is a fundamental similarity measure used in income-inequality analysis, cluster validation, reliability engineering, and goodness-of-fit testing. In many practical settings, the integral admits no closed form, so Monte Carlo evaluation is required, and variance reduction is of direct computational interest. This paper develops an asymptotic framework for regression-adjusted Monte Carlo estimation of when the auxiliary control variate is an estimated density ratio, and characterises the exact conditions under which the adjustment delivers genuine variance reduction. Methodologically, sample splitting separates density estimation from Monte Carlo integration, making the integrand summands conditionally i.i.d. and enabling a law-of-total-variance decomposition that isolates Monte Carlo noise from density-estimation error. Two estimators are studied: a centering-at-1 estimator and a bias-corrected estimator that estimates the centering mean from an independent subsample. For the centering-at-1 estimator, we establish consistency and derive the unconditional asymptotic variance; under an equal split, the density-estimation penalty can exceed the variance-reduction gain. For the bias-corrected estimator we derive the exact conditional variance and the optimal main/centering split; measured against plain Monte Carlo at the same main-sample budget, the optimal split delivers strict variance reduction if and only if , with the oracle gain recovered as the centering sample dominates. Simulations across normal, non-normal, and multivariate designs confirm these predictions. The framework extends to mixture models and nonparametric overlap estimation under standard regularity conditions, contributing both to the literature on regression-adjusted simulation and to overlap-based statistical inference in applied sciences.