发表机构
University of Turin; Collegio Carlo Alberto; Bocconi University; University of Notre Dame（都灵大学; 卡洛阿尔贝托学院; 博科尼大学; 圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对泊松混合模型中随机变量之和的估计问题，提出基于最小距离的非参数经验贝叶斯方法，实现渐近最优并给出有限样本保证，数值实验验证性能。
AI 中文摘要
可观测与不可观测变量函数之和的估计是统计学中长期存在的问题，在许多领域都有应用。我们在泊松混合模型中考虑这一问题，其中经验贝叶斯提供了自然的框架，但非参数理论仍然有限。我们开发了一种基于正则和粗化最小距离估计未知混合分布的非参数经验贝叶斯方法。对于此类和的一大类，我们建立了大样本保证，表明所得的插件估计渐近地合并到Oracle贝叶斯估计。特别地，当混合分布具有有限支撑时，我们获得了接近参数的收敛速率，仅差对数因子。然后，我们提供了两个代表性和的有限样本分析：观测计数不超过固定阈值的单元的总强度，以及观测计数超过其潜在强度的单元的数量。对于总强度和，我们建立了极小极大后悔的下界，并在混合分布的紧支撑和次指数假设下，推导了正则和粗化最小距离程序的上界。这些上界与极小极大速率匹配至对数因子，在紧支撑设置中，当粗化水平固定时，粗化程序恰好匹配该速率。高于平均值的和表现出显著不同的行为：极小极大后悔的下界表明，有界后悔通常不可能实现。我们确定了额外的稳定性条件，在这些条件下，有界后悔可以恢复至对数因子，并表明当混合分布具有有限支撑时，这些条件自动满足。在合成和真实数据上的数值实验说明了所提出方法的性能。
英文摘要
The estimation of sums of functions of observable and unobservable variables is a long-standing problem in statistics, with applications in many domains. We consider this problem in Poisson mixture models, where empirical Bayes provides a natural framework but nonparametric theory remains limited. We develop a nonparametric empirical Bayes methodology based on regular and coarsened minimum-distance estimation of the unknown mixing distribution. For a broad class of such sums, we establish large-sample guarantees showing that the resulting plug-in estimates asymptotically merge with the oracle Bayes estimate. In particular, when the mixing distribution has finite support, we obtain a nearly parametric convergence rate, up to a logarithmic factor. We then provide a finite-sample analysis of two representative sums: the total intensity among units whose observed count does not exceed a fixed threshold, and the number of units whose observed count exceeds their latent intensity. For the total-intensity sum, we establish a lower bound on the minimax regret and derive upper bounds for both regular and coarsened minimum-distance procedures under compact-support and subexponential assumptions on the mixing distribution. These upper bounds match the minimax rate up to logarithmic factors, with the coarsened procedure in the compact-support setting matching the rate exactly when the coarsening level is fixed. The above-average sum displays a markedly different behavior: a lower bound on the minimax regret shows that bounded regret is in general impossible to achieve. We identify additional stability conditions under which bounded regret can be recovered up to logarithmic factors, and show that these conditions are automatically satisfied when the mixing distribution has finite support. Numerical experiments on both synthetic and real data illustrate the performance of the proposed methodology.
Comments104 pages, 22 figures
