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.

翻译：本研究聚焦于形如$X=\xi_1+\ldots+\xi_N$的复合模型的统计推断问题，其中$N$为表示加项个数的随机变量，加项$\xi_1, \xi_2, \ldots$为独立同分布(i.i.d.)随机变量。论文探讨了在已知$N$分布的前提下，通过观测$X$分布样本重建$\xi$分布的问题——该过程称为解复合。本工作突破了传统研究将$N$分布限定为泊松类型的框架，从而拓展至更广泛的研究背景。我们提出了$\xi$密度的非参数估计量，推导了其收敛速度，并证明在$\xi$与$N$的适当分布类别中该速度达到极小极大最优。最后，通过模拟算例展示了该算法的数值性能。