Nonparametric estimation of sliced inverse regression by the $ k$-nearest neighbors kernel method

TL;DR

This work develops a nonparametric framework for estimating the sliced inverse regression (SIR) EDR space using a $k$-nearest neighbors kernel approach. By nonparametrically estimating the density $f$ and the regression components $g_\\ell$, the authors construct $\\widehat{\\Lambda}_n$ from which the EDR directions follow via spectral decomposition, with a random bandwidth determined by $k_n$ nearest neighbors. They establish consistency and asymptotic normality for $\\widehat{\\Lambda}_n$ and its eigenvectors under a comprehensive set of regularity conditions, and they validate the method through simulations that compare favorably with the classical Zhu (kernel) SIR. The results indicate that the $k$-NN kernel approach provides a bandwidth-availability advantage while delivering competitive finite-sample performance for dimension reduction in regression.

Abstract

We investigate nonparametric estimation of sliced inverse regression (SIR) via the $k$-nearest neighbors approach with a kernel. An estimator of the covariance matrix of the conditional expectation of the explanatory random vector given the response is then introduced, thereby allowing to estimate the effective dimension reduction (EDR) space. Consistency of the proposed estimators is proved through derivation of asymptotic normality. A simulation study, made in order to assess the finite-sample behaviour of the proposed method and to compare it to the kernel estimate, is presented.

Theorems & Definitions (13)

• Remark 2.1
• Remark 3.1
• Theorem 3.1
• Theorem 3.2
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• Lemma 5.4
• Lemma 5.5
• Lemma 5.6