Minimal penalty for Goldenshluger-Lepski method

This paper is concerned with adaptive nonparametric estimation using the Goldenshluger-Lepski methodology. This method is designed to select an estimator among a collection $(\hat f\_h)\_{h\in H}$ by minimizing $B(h)+V(h)$ with $B(h)=\sup\{[\|\hat f\_{h'}- \hat f\_{h}\|-V(h')]\_+, \: h'\in H\}$ and $V(h)$ a variance term. In the case of density estimation with kernel estimators, it is shown that the procedure fails if the variance term is chosen too small: this gives for the first time a minimal penalty for the Goldenshluger-Lepski methodology. Some simulations illustrate the theoretical results.