AI 中文总结
研究 M/G/1 工作负载模型中服务时间分布的非参数推断问题，提出基于工作负载轨迹的估计器 $B_n(w)$，通过筛选机制简化问题，在适度正则性假设下建立误差界，给出首个达参数化 $L^1$ 风险率且无需相关假设的解决方案。
AI 中文摘要
我们解决了 Hansen 和 Pitts（2006 年）提出的关于 M/G/1 工作负载模型中服务时间分布的非参数推断这一长期存在的开放性问题。考虑一个到达率 $\lambda>0$ 和服务时间分布 $B(\cdot)$ 未知的 M/G/1 队列，不假设稳定性或平稳性。统计学家在离散时间 $t = 0,1,\ldots,n$ 观察工作负载过程，旨在估计固定点 $w>0$ 处的 $B(w)$。我们提出了仅基于观察到的工作负载轨迹的估计器 $B_n(w)$。其构建依赖于一种筛选机制，该机制从工作负载过程中提取条件独立同分布的复合泊松增量，从而将相关数据问题简化为拉普拉斯变换反演框架。在对 $B(\cdot)$ 的适度正则性假设下，即 $[0,\infty)$ 上连续可微、在 $w$ 处二阶可微且二阶矩有限，我们建立了界 \(\mathbb{E}\bigl|B_n(w)-B(w)\bigr| =\mathcal{O}\!\left(\frac{\log n}{\sqrt{n}}\right)\)，\(n\to\infty\)。这为 Hansen - Pitts 问题提供了第一个实现参数化 $L^1$ 风险率（直至对数因子）的解决方案，无需平稳性、稳定性或到达率知识。
英文摘要
Hansen and Pitts (2006) introduced the problem of nonparametric estimation of the service-time distribution of an M/G/1 queue observed through its workload process at discrete times $t=0,1,\ldots,n$. Despite its seemingly simple formulation, obtaining an estimator with sharp risk guarantees for this observation model has remained an open challenge for nearly two decades. In this paper, we construct such an estimator and prove that its $L^1$-risk is $\mathcal{O}\left(\frac{\log n}{\sqrt{n}}\right)$ as $n\to\infty$. Remarkably, this nearly parametric convergence rate is achieved without assuming stationarity, stability, or knowledge of the arrival rate. Our approach is based on a two-stage screening procedure that uncovers a hidden conditionally independent compound Poisson structure within the dependent workload observations. This probabilistic reduction transforms the original estimation problem into a classical decompounding problem, making it possible to leverage existing nonparametric estimation techniques despite the complex dependence induced by the reflected workload process. More broadly, we hope that the proposed screening methodology will provide a useful framework for statistical inference from dependent stochastic systems beyond the classical assumption of stability.
