Nonparametric adaptive estimation for grouped data

•Constructive nonparametric adaptive density estimator for grouped data.•Definition of the logarithm of the empirical characteristic function.•Upper and lower bound results that ensure optimality.

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.