Adaptive non-asymptotic confidence balls in density estimation

Abstract

Abstract:We build confidence balls for the common density s of a real valued sample X 1 , ..., X n . We use resampling methods to estimate the projection of s onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all n ≥ 2 and the balls are adaptive over a collection of linear spaces.

A review of uncertainty quantification for density estimation

“…Although they acknowledged the possibility of replacing some of the unknown terms in their construction by certain data-driven approximations, they did not consider applications to real data. Lerasle [122] provided a different approach to L 2 confidence balls, the full intricacies of which are omitted here. They used a model selection approach to determine the best approximation space (they dealt more generally with projection estimators on linear subspaces of an L 2 space, but for our purposes it suffices to consider the special case of wavelet estimators where the selection is for the truncation level) and a resampling method to estimate an L 2 norm needed in the radius of the set, thereby avoiding the sample splitting needed by some of the other literature discussed here.…”