Strong consistency of kernel estimator in a semiparametric regression model

Abstract

ABSTRACTEstimating the effective dimension reduction (EDR) space, related to the semiparametric regression model introduced by Li [Sliced inverse regression for dimension reduction. J Amer Statist Assoc. 1991;86:316–327], is based on the estimation of the covariance matrix Λ of the conditional expectation of the vector of predictors given the response. An estimator of Λ based on kernel method was introduced by Zhu and Fang [Asymptotics for kernel estimate of sliced inverse regression. Ann Statist. 1996;24:1053–1068] who then derived, under some conditions, the asymptotic distribution of , as . In this paper, we obtain the almost sure convergence of to Λ, as .