On Adaptive Wavelet Estimation of a Class of Weighted Densities

Abstract

We investigate the estimation of a weighted density taking the form g = w(F)f, where f denotes an unknown density, F the associated distribution function and w is a known non-negative weight. Such a class encompasses many examples, including those arising in order statistics or when g is related to the maximum or the minimum of N (random or fixed) independent and identically distributed (i.i.d.) random variables. We here construct a new adaptive non-parametric estimator for g based on a plug-in approach and the wavelets methodology. For a wide class of models, we show that it attains fast rates of convergence under the risk with p ⩾ 2 over Besov balls. Our estimator is also simple to implement and fast. We also report an extensive simulation study to support our findings.