Decompounding Under General Mixing Distributions

TL;DR

This paper develops a nonparametric method for decompounding in compound models with general mixing distributions, providing optimal convergence rates and demonstrating numerical effectiveness on simulated data.

Contribution

It introduces a novel nonparametric estimator for the summand distribution in compound models with arbitrary mixing distributions, extending beyond the Poisson case.

Findings

Estimator achieves minimax optimal convergence rates

Method performs well on simulated examples

Broadens decompounding applicability to general mixing distributions

Abstract

This study focuses on statistical inference for compound models of the form $X=\xi_1+\ldots+\xi_N$, where $N$ is a random variable denoting the count of summands, which are independent and identically distributed (i.i.d.) random variables $\xi_1, \xi_2, \ldots$. The paper addresses the problem of reconstructing the distribution of $\xi$ from observed samples of $X$'s distribution, a process referred to as decompounding, with the assumption that $N$'s distribution is known. This work diverges from the conventional scope by not limiting $N$'s distribution to the Poisson type, thus embracing a broader context. We propose a nonparametric estimate for the density of $\xi$, derive its rates of convergence and prove that these rates are minimax optimal for suitable classes of distributions for $\xi$ and $N$. Finally, we illustrate the numerical performance of the algorithm on simulated examples.