Efficient estimation of conditional covariance matrices for dimension reduction

Let X∈Rp$\boldsymbol{X}\in \mathbb {R}^p$ and Y∈R$Y\in \mathbb {R}$. In this paper, we propose an estimator of the conditional covariance matrix, Cov(E[X|Y])$\operatorname{Cov}(\mathbb {E}[\boldsymbol{X}\vert Y])$, 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])$\operatorname{Cov}(\mathbb {E}[\boldsymbol{X}\vert Y])$ 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.