Sequential Data-Adaptive Bandwidth Selection by Cross-Validation for Nonparametric Prediction

TL;DR

This paper develops a sequential cross-validation method for selecting bandwidths in nonparametric regression, enabling simultaneous estimation, prediction, and change detection with proven consistency under dependent errors.

Contribution

It introduces a unified sequential bandwidth selection approach using cross-validation that works for dependent time series data, with theoretical guarantees.

Findings

Proves uniform weak laws of large numbers for the cross-validation statistic.

Establishes weak consistency of the cross-validated bandwidth in dependent data.

Demonstrates the method's effectiveness on photovoltaic data.

Abstract

We consider the problem of bandwidth selection by cross-validation from a sequential point of view in a nonparametric regression model. Having in mind that in applications one often aims at estimation, prediction and change detection simultaneously, we investigate that approach for sequential kernel smoothers in order to base these tasks on a single statistic. We provide uniform weak laws of large numbers and weak consistency results for the cross-validated bandwidth. Extensions to weakly dependent error terms are discussed as well. The errors may be {\alpha}-mixing or L2-near epoch dependent, which guarantees that the uniform convergence of the cross validation sum and the consistency of the cross-validated bandwidth hold true for a large class of time series. The method is illustrated by analyzing photovoltaic data.