Nonparametric adaptive estimation for grouped data

TL;DR

This paper introduces a nonparametric adaptive method for estimating the density of a random variable from grouped data, specifically sums of independent copies, with proven minimax optimality and practical numerical validation.

Contribution

It presents a novel adaptive estimator based on the empirical characteristic function for grouped data, extending applicability and achieving near-optimal performance.

Findings

Estimator is minimax-optimal up to logarithmic factors

Numerical experiments demonstrate effective performance

Method applies to broader contexts beyond the studied setting

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 $\ge$ 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.