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Suppose that P is the distribution of a pair of random variables (X, Y) on a product space X x V with known marginal distributions Px and Py. We study efficient estimation of functions 0(h) = fh dP for fixed h: X x Y -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 X and \V 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 L2(P), or equivalently by a pair of equations which we call the "ACE equations."

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