On Asymptotic Normality of the Local Polynomial Regression Estimator with Stochastic Bandwidths

TL;DR

This paper proves that local polynomial regression estimators with data-driven, stochastic bandwidths are asymptotically equivalent to those with fixed, nonstochastic bandwidths, extending previous theoretical results.

Contribution

It establishes the asymptotic equivalence in probability between stochastic and nonstochastic bandwidth local polynomial estimators, broadening theoretical understanding.

Findings

Establishes asymptotic equivalence of estimators with stochastic and fixed bandwidths

Extends previous work by Boente and Fraiman (1995) and Ziegler (2004)

Provides theoretical foundation for data-driven bandwidth selection methods

Abstract

Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are nonstochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result nonparametric estimators are usually constructed using stochastic bandwidths. In this paper we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstochastic bandwidths. Our result extends previous work by Boente and Fraiman (1995) and Ziegler (2004).