The Weitzman overlap coefficient Δ =
min{f1, f2} 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 Δ 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 biascorrected
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 |ρk| > 1/
√
2, with the oracle
gain 1 − ρ2
k 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.