Minimal penalty for Goldenshluger-Lepski method

TL;DR

This paper investigates the Goldenshluger-Lepski method for adaptive density estimation, revealing a phase transition at a minimal penalty level where the estimator's risk dramatically worsens if the penalty is too small.

Contribution

It demonstrates for the first time a minimal penalty threshold in the Goldenshluger-Lepski method, showing the risk explosion below this critical penalty in density estimation.

Findings

Risk explodes when penalty is below critical value

Risk remains controlled above the critical penalty

Simulations support theoretical phase transition results

Abstract

This paper is concerned with adaptive nonparametric estimation using the Goldenshluger-Lepski selection method. This estimator selection method is based on pairwise comparisons between estimators with respect to some loss function. The method also involves a penalty term that typically needs to be large enough in order that the method works (in the sense that one can prove some oracle type inequality for the selected estimator). In the case of density estimation with kernel estimators and a quadratic loss, we show that the procedure fails if the penalty term is chosen smaller than some critical value for the penalty: the minimal penalty. More precisely we show that the quadratic risk of the selected estimator explodes when the penalty is below this critical value while it stays under control when the penalty is above this critical value. This kind of phase transition phenomenon for penalty calibration has already been observed and proved for penalized model selection methods in various contexts but appears here for the first time for the Goldenshluger-Lepski pairwise comparison method. Some simulations illustrate the theoretical results and lead to some hints on how to use the theory to calibrate the method in practice.