Efficient estimation of conditional covariance matrices for dimension reduction

Abstract : Let X ∈ Rp andY ∈ R. In this paper,we propose an estimator of the conditional covariancematrix, 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.