Theoretical Properties and Practical Performance of Fully Robust One-Sided Cross-Validation

TL;DR

This paper examines the theoretical and practical aspects of fully robust one-sided cross-validation (OSCV), comparing different kernels and highlighting issues like sensitivity and multiple minima, to improve bandwidth selection in regression.

Contribution

It provides a comparative analysis of $H_I$- and $\

Findings

$H_I$ kernel tends to produce too low bandwidths in smooth cases.

$H_I$-based OSCV curves exhibit wiggles near zero.

Robust bimodal kernels often lead to multiple local minima in OSCV curves.

Abstract

Fully robust OSCV is a modification of the OSCV method that produces consistent bandwidth in the cases of smooth and nonsmooth regression functions. The current implementation of the method uses the kernel $H_I$ that is almost indistinguishable from the Gaussian kernel on the interval $[-4,4]$, but has negative tails. The theoretical properties and practical performances of the $H_I$- and $\phi$-based OSCV versions are compared. The kernel $H_I$ tends to produce too low bandwidths in the smooth case. The $H_I$-based OSCV curves are shown to have wiggles appearing in the neighborhood of zero. The kernel $H_I$ uncovers sensitivity of the OSCV method to a tiny modification of the kernel used for the cross-validation purposes. The recently found robust bimodal kernels tend to produce OSCV curves with multiple local minima. The problem of finding a robust unimodal nonnegative kernel remains open.