Efficient estimation of conditional covariance matrices for dimension reduction

TL;DR

This paper introduces an efficient semi-parametric estimator for the conditional covariance matrix in inverse regression, improving dimension reduction techniques by leveraging quadratic functional estimation and asymptotic analysis.

Contribution

It proposes a novel estimator for the conditional covariance matrix based on quadratic functional estimation, with theoretical analysis of its asymptotic properties.

Findings

Estimator is asymptotically efficient.

Method effectively estimates conditional covariance in high dimensions.

Theoretical properties are rigorously established.

Abstract

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