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

TL;DR

This paper introduces a nonparametric method using k-nearest neighbors and kernel techniques to estimate the sliced inverse regression (SIR), enabling effective dimension reduction with proven consistency and asymptotic normality.

Contribution

It proposes a novel nonparametric estimator for SIR based on k-nearest neighbors and kernel methods, with theoretical guarantees and empirical evaluation.

Findings

Estimator is consistent and asymptotically normal.

Simulation shows good finite-sample performance.

Method compares favorably to existing kernel estimates.

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.