Density estimation with quadratic loss: a confidence intervals method

¹ Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, France; alquier@ensae.fr
² Laboratoire de Statistique, CREST 3, avenue Pierre Larousse, 92240 Malakoff, France.

Received: 10 October 2007

Abstract

We propose a feature selection method for density estimation with quadratic loss. This method relies on the study of unidimensional approximation models and on the definition of confidence regions for the density thanks to these models. It is quite general and includes cases of interest like detection of relevant wavelets coefficients or selection of support vectors in SVM. In the general case, we prove that every selected feature actually improves the performance of the estimator. In the case where features are defined by wavelets, we prove that this method is adaptative near minimax (up to a log term) in some Besov spaces. We end the paper by simulations indicating that it must be possible to extend the adaptation result to other features.