Regression Smoothing Parameters that are not Far from their Optimum

Abstract

It is well known that data-driven regression smoothing parameters ħ based on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow as n ^–1/10; that is, for a bandwidth ħ ₀ optimizing the averaged squared error, n ^1/10 (ħ — ħ ₀)/ħ ₀ tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidths h ₀. This (nonrandom) smoothing parameter can be approximated much faster. We use the technique of double smoothing to show that there is an ħ such that, under certain conditions, n ^1/2(ħ − h ₀)/h ₀ tends to an asymptotic normal distribution.

Key Words:

• Automatic smoothing
• Double smoothing
• Kernel estimation
• Nonparametric regression