Asymptotic minimax risk of predictive density estimation for non-parametric regression

TL;DR

This paper analyzes the asymptotic minimax risk in predictive density estimation within non-parametric regression models with Gaussian errors, establishing convergence rates and constants for Bayesian and all estimators.

Contribution

It derives the exact asymptotics of the minimax risk under Kullback-Leibler divergence for Gaussian errors in non-parametric regression.

Findings

Minimax risk convergence rate is established.

Bayesian predictive densities achieve asymptotic minimax risk.

Minimax risk among Bayesian and all estimators are asymptotically equivalent.

Abstract

We consider the problem of estimating the predictive density of future observations from a non-parametric regression model. The density estimators are evaluated under Kullback--Leibler divergence and our focus is on establishing the exact asymptotics of minimax risk in the case of Gaussian errors. We derive the convergence rate and constant for minimax risk among Bayesian predictive densities under Gaussian priors and we show that this minimax risk is asymptotically equivalent to that among all density estimators.