On the estimation of the order of smoothness of the regression function

TL;DR

This paper introduces a Bayesian approach to estimate the smoothness order of regression functions, focusing on Bernstein polynomial estimates, offering faster computation and reliable uncertainty quantification compared to traditional cross-validation methods.

Contribution

The paper proposes a novel Bayesian method for selecting the polynomial degree in nonparametric regression, demonstrating asymptotic optimality and computational efficiency.

Findings

Method is one or two orders of magnitude faster than cross-validation.

Provides comparable predictive accuracy to cross-validation.

Enables simultaneous quantification of model uncertainty and parameter estimates.

Abstract

The order of smoothness chosen in nonparametric estimation problems is critical. This choice balances the tradeoff between model parsimony and data overfitting. The most common approach used in this context is cross-validation. However, cross-validation is computationally time consuming and often precludes valid post-selection inference without further considerations. With this in mind, borrowing elements from the objective Bayesian variable selection literature, we propose an approach to select the degree of a polynomial basis. Although the method can be extended to most series-based smoothers, we focus on estimates arising from Bernstein polynomials for the regression function, using mixtures of g-priors on the model parameter space and a hierarchical specification for the priors on the order of smoothness. We prove the asymptotic predictive optimality for the method, and through simulation experiments, demonstrate that, compared to cross-validation, our approach is one or two orders of magnitude faster and yields comparable predictive accuracy. Moreover, our method provides simultaneous quantification of model uncertainty and parameter estimates. We illustrate the method with real applications for continuous and binary responses.