Optimal adaptive estimation of the relative density

• Gaëlle Chagny ^[1] ; Claire Lacour ^[2]
  1. [1] University of Rouen

     University of Rouen

     Arrondissement de Rouen, Francia

     
  2. [2] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
• Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 24, Nº. 3, 2015, págs. 605-631
• Idioma: inglés
• DOI: 10.1007/s11749-015-0426-6
• Texto completo no disponible (Saber más ...)
• Resumen

  • This paper deals with the classical statistical problem of comparing the probability distributions of two real random variables X and X0, from a double independent sample. While most of the usual tools are based on the cumulative distribution functions F and F0 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 F0(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 preliminary estimator of F0. 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.