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Title: Comparison methods for bandwidth selection Authors:  Claire Lacour - Paris-Est University / INRIA (France) [presenting]
Pascal Massart - University Paris-Sud (France)
Vincent Rivoirard - Paris Dauphine University (France)
Suzanne Varet - Paris-Sud University (France)
Abstract: The problem of estimating a density with kernel estimators is considered. A classical issue is the choice of the bandwidth. We focus on the Goldenshluger-Lepski selection method, which is based on pairwise comparisons between estimators with respect to some loss function. The method also involves a penalty term than typically needs to be large enough in order that the method works (in the sense that one can prove some oracle type inequality for the selected estimator). In the case of the quadratic loss, we study the procedure for different values of the tuning parameter. We give a minimal value of the penalty, beyond which the procedure fails, that brings to light a phase transition phenomenon for penalty calibration. Moreover, we highlight a degenerate case, where all the estimators are compared to the overfitted one. This leads to a new method which is in some sense intermediate between Lepski's method and penalized empirical risk minimization. We provide some theoretical results which lead to some fully data-driven selection strategy. We will also show the numerical performance of the method.