Decompounding Under General Mixing Distributions

This study focuses on statistical inference for compound models of the form X=ξ_1+…+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, …. 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.