On efficient estimation of linear functionals of a bivariate distribution with known marginals

In this paper we construct efficient estimators for linear functionals of a bivariate distribution with known marginals. Previously, Bickel et al. (Ann. Statist. 19 (1991) 1316) constructed such estimators using the modified minimum chi-square principle. Our estimators utilize the least-squares principle and orthonormal bases for the Hilbert spaces of square integrable functions under the known marginal distributions and are easy to compute. Simulations indicate that in the moderate sample sizes considered our estimator compares favorably with the one proposed by Bickel et al.