The Poisson compound decision problem is a long-standing problem is statistics, for which empirical Bayes methods are commonly used to estimate Poisson means in static or batch settings. We consider this problem in a streaming, or online, framework. Building on a quasi-Bayesian approach based on Newton's algorithm, we develop a sequential estimate that is easy to evaluate, computationally efficient, and has constant per-observation cost as the data accrue. We establish frequentist guarantees for the proposed estimate, including consistency and asymptotic optimality, with optimality understood as vanishing excess Bayes risk, or regret. Empirical performance is assessed through simulation studies and comparisons with benchmark procedures.

翻译：泊松复合决策问题是统计学中一个长期存在的经典问题，通常采用经验贝叶斯方法在静态或批量场景下估计泊松均值。我们考虑该问题在流式或在线框架下的求解方案。基于牛顿算法的准贝叶斯方法，我们构建了一种序列估计量，该估计量易于计算、具有高效性，且随着数据累积其单次观测的计算成本保持恒定。我们为所提出的估计量建立了频率学派保证，包括一致性和渐近最优性，其中最优性通过消失的超额贝叶斯风险（即遗憾值）来定义。通过模拟研究及与基准方法的对比，我们评估了该方法的实证性能。