Adaptive quadratic functional estimation of a weighted density by model selection

Abstract

We consider the problem of estimating the integral of the square of a probability density function f on the basis of a random sample from a weighted distribution. Specifically, using model selection via a penalized criterion, an adaptive estimator for ∫ f 2 based on weighted data is proposed for probability density functions which are uniformly bounded and belong to certain Besov bodies. We show that the proposed estimator attains the minimax rate of convergence that is optimal in the case of direct data. Additionally, we obtain the information bound for the problem of estimating ∫ f 2 when weighted data are available and compare it with the information bound for the case of direct data. A small simulation study is conducted to illustrate the usefulness of the proposed estimator in finite sample situations.