Decompounding Under General Mixing Distributions

Abstract

This study focuses on statistical inference for compound models of the form $X=ξ_1+\ldots+ξ_N$, where $N$ is a random variable denoting the count of summands, which are independent and identically distributed (i.i.d.) random variables $ξ_1, ξ_2, \ldots$. The paper addresses the problem of reconstructing the distribution of $ξ$ 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 $ξ$, derive its rates of convergence and prove that these rates are minimax optimal for suitable classes of distributions for $ξ$ and $N$. Finally, we illustrate the numerical performance of the algorithm on simulated examples.