Efficient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the least-squares approach

In this paper, we characterize and construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. An efficient estimator equals the empirical estimator minus a correction term and provides significant improvements over the empirical estimator. We construct an efficient estimator by estimating the correction term. For this we use the least-squares principle and an estimated orthonormal basis for the Hilbert space of square-integrable functions under the unknown equal marginal distribution. Simulations confirm the asymptotic behavior of this estimator in moderate sample sizes and the considerable theoretical gains over the empirical estimator.