Efficient Estimation of Linear Functionals of a Probability Measure $P$ with Known Marginal Distributions

Abstract

Suppose that $P$ is the distribution of a pair of random variables $(X, Y)$ on a product space $\mathbb{X} \times \mathbb{Y}$ with known marginal distributions $P_X$ and $P_Y$. We study efficient estimation of functions $\theta(h) = \int h dP$ for fixed $h: \mathbb{X} \times \mathbb{Y} \rightarrow R$ under iid sampling of $(X, Y)$ pairs from $P$ and a regularity condition on $P$. Our proposed estimator is based on partitions of both $\mathbb{X}$ and $\mathbb{Y}$ and the modified minimum chi-square estimates of Deming and Stephan (1940). The asymptotic behavior of our estimator is governed by the projection on a certain sum subspace of $L_2(P)$, or equivalently by a pair of equations which we call the "ACE equations."