Optimal adaptive estimation of the relative density

Gaëlle Chagny (1) , Claire Lacour (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

This paper deals with the classical statistical problem of comparing the probability distributions of two real random variables $X$ and $X_0$, from a double independent sample. While most of the usual tools are based on the cumulative distribution function $F$ and $F_0$ of the variables, we focus on the relative density, a function recently used in two-sample problems, and defined as the density of the variable $F_0(X)$. We provide a nonparametric adaptive strategy to estimate the target function. We first define a collection of estimates using a projection on the trigonometric basis and a preliminar estimator of $F_0$. An estimator is selected among this collection of projection estimates, with a criterion in the spirit of the Goldenshluger-Lepski methodology. We show the optimality of the procedure both in the oracle and the minimax sense: the convergence rate for the risk computed from an oracle inequality matches with the lower bound, that we also derived. Finally, some simulations illustrate the method.

Mots clés: two-sample problem; relative density; model selection; Nonparametric estimation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Gaëlle Chagny, Claire Lacour. Optimal adaptive estimation of the relative density. Test, 2015, 24 (3), pp.605-631. ⟨10.1007/s11749-015-0426-6⟩. ⟨hal-00955161v2⟩