Strong consistency of kernel method for sliced average variance estimation

Abstract

In this paper, we consider the kernel method to estimate sliced average variance estimation (SAVE). SAVE is a tool as SIR (sliced inverse regression), recommended to identify and to estimate the central dimension reduction (CDR) subspace. CDR subspace is the intersection of all dimension reduction subspaces which are at the base to describe the conditional distribution of the response Y given a $d-$ dimensional predictor vector X. SAVE and even SIR are used to estimate CDR subspace. Two versions are very popular: slice version and kernel version. In this paper, we are looking at the kernel version. For Kernel SAVE version, two asymptotic properties have been demonstrated in particular: asymptotic normality and convergence in probability. And we know that these two properties, although important, are weak in front of almost sure convergence. However, until now, the strong consistency has not yet been obtained. In this paper, we obtain, under weaker assumptions, this asymptotic property.

Keywords:

• Strong consistency
• dimension reduction
• kernel estimator
• semiparametric regression
• sliced average variance estimation

Acknowledgment

The author is very grateful to an anonymous reviewer for his valuable comments which helped to improve the article.