Nonparametric adaptive estimation for grouped data

Abstract

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.

Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

“…Comments on the adaptive procedure. In the present paper we develop an adaptive procedure that was successfully used in of Duval and Kappus [26] which considers the problem of grouped data estimation. One observes i.i.d.…”

“…Both in the grouped data setting (see [26]) and in the deconvolution setting (see Section 2) the computation of the adaptive cutoff, after simplifications, involves the set m n ∈ | ϕ Y,n (u)| = 1 √ n (1 + κ log n) , κ > 0 (4.1)…”

“…Suppose Y is related to a variable X, with Lebesgue-density f through a known transformation T relating their characteristic functions: ϕ Y = T(ϕ X ), where T can be linear (e.g. deconvolution) or not (grouped data (see [46,26]), decompounding). We are interested in estimating the density f of X from the (Y j ).…”

“…Proof of Theorem 3.1 uses similar arguments as the proof of Theorem 1 of Duval and Kappus [26]. However, estimator (3.4) is different from the estimator studied in [26] and we need to take into account the additional parameter ∆ that needs to be carefully handled.…”

“…In the present paper, we put the stress on an adaptive method that was successfully applied in Duval and Kappus [26] and whose scope goes beyond the grouped data setting studied therein. The adaptive procedure presented below does not outperform existing techniques, it suffers a logarithmic loss, but has the advantage of being numerically simple and fast and permits to compute the optimal cutoff parameter for different classical inverse problems.…”