Statistical estimation aims at building procedures to recover unknown parameters by nalysing some measured data sampled from a large population. This note deals with the case of infinite dimensional parameters, that is functions, through the example of probability density estimation. After discussing how to quantify the performances of estimation methods, we discuss the limits of accuracy of any estimator for the density (minimax point of view) and present the main two methods of nonparametric estimation: projection and kernel estimators. Upper-bounds on the accuracy of the defined estimators for a fixed amount of data are derived. They highly depend on smoothing parameters (the model dimension and the bandwidth, respectively for the two methods), which should be carefully chosen. The second part of the text is devoted to data-driven estimator selection, for which we provide a brief review: both the model selection and the bandwidth choice issues are addressed. We describe two methods that permit to obtain so-called oracle-type inequalities while being adaptive: the selection does not depend on the unknown smoothness of the target density. A large list of references is provided, and numerical experiments illustrate the theoretical results.
Received: July 2016
Published online: December 2016
Gaëlle ChagnyLaboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen Normandie, France.