Efficient estimation of conditional covariance matrices for dimension reduction

Abstract

Let X∈Rp and Y∈R. In this paper, we propose an estimator of the conditional covariance matrix, Cov(E[X|Y]), in an inverse regression setting. Based on the estimation of a quadratic functional, this methodology provides an efficient estimator from a semi parametric point of view. We consider a functional Taylor expansion of Cov(E[X|Y]) under some mild conditions and the effect of using an estimate of the unknown joint distribution. The asymptotic properties of this estimator are also provided.