Nonparametric adaptive estimation for grouped data

The aim of this paper is to estimate the density f of a random variable X when one has access to independent observations of the sum of K ≥ 2 independent copies of X. We provide a constructive estimator based on a suitable definition of the logarithm of the empirical characteristic function. We propose a new strategy for the data driven choice of the cut-off parameter. The adaptive estimator is proven to be minimax-optimal up to some logarithmic loss. A numerical study illustrates the performances of the method. Moreover, we discuss the fact that the definition of the estimator applies in a wider context than the one considered here.