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 [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 ${\hat{\mathrm{\Lambda }}}_n$ 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 $\sqrt{n}({\hat{\mathrm{\Lambda }}}_n-\mathrm{\Lambda })$, as $n\to +\mathrm{\infty }$. In this paper, we obtain the almost sure convergence of ${\hat{\mathrm{\Lambda }}}_n$ to Λ, as $n\to +\mathrm{\infty }$.

KEYWORDS:

• Strong consistency
• Kernel estimator
• semiparametric
• regression

Acknowledgements

The authors are very grateful to an anonymous referee for his valuable comments which permited to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.