Strong consistency of kernel estimator in a semiparametric regression model

Abstract

Estimating the effective dimension reduction (EDR) space, related to the semiparametric regression model introduced by Li \cite{sir}, is based on the estimation of the covariance matrix $\Lambda$ of the conditional expectation of the vector of predictors given the response. An estimator $\widehat{\Lambda}_n$ of $\Lambda $ based on kernel method was introduced by Zhu and Fang \cite{Asymptotics} who then derived, under some conditions, the asymptotic distribution of $\sqrt{n}\left(\widehat{\Lambda}_n-\Lambda\right)$, as $n\rightarrow +\infty$. In this paper, we obtain, under specified conditions, the almost sure convergence of $\widehat{\Lambda}_n$ to $\Lambda$, as $n\rightarrow +\infty$.