似然比分析方法向Skorokhod $M_1$拓扑的推广（应用于泊松平滑变点模型）
Extension of Likelihood Ratio Analysis Method to Skorokhod $M_1$ Topology (with Application to Poissonian Smooth Change-Point Model)
- Univ. Lille（里尔大学）
- CNRS, UMR 8524 — Laboratoire Paul Painlevé（法国国家科学研究中心，联合研究实验室8524——保罗·潘勒韦实验室）
- Seenovate
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
将似然比分析方法推广至Skorokhod $M_1$拓扑，处理极限似然比过程不连续的模型，并应用于快速机制下的泊松平滑变点模型，证明极大似然估计量具有与纯变点模型相同的渐近行为。
AI中文摘要：
我们将一维参数情形下的Ibragimov-Khasminskii似然比分析方法推广到基于Skorokhod $M_1$拓扑的框架中。该框架适用于一大类统计模型，包括那些归一化似然比过程连续而极限似然比过程不连续的模型，这一情形超出了基于一致拓扑或Skorokhod $J_1$拓扑的经典方法的适用范围。我们在定义于$\RR$上、在$\pm\infty$处消失的càdlàg函数空间（赋予Skorokhod $M_1$拓扑）中，推导了似然比过程弱收敛的充分条件，并展示了该收敛如何导出极大似然估计量的渐近行为。我们还引入了控制$M_1$连续模的新技术。尽管这些技术特别适用于所有跳跃和陡峭连续过渡都朝同一方向的模型，但它们具有独立的研究价值，并可能在此类情形之外发挥作用。作为应用，我们研究了快速机制下非齐次泊松过程的平滑变点模型，其中过渡区间收缩速度快于$1/n$。我们证明了在这种情况下，极大似然估计量与相应的“纯”变点模型具有相同的渐近行为（相合性、收敛速率、极限分布和矩收敛）。
英文摘要：
We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod $M\_1$ topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous while the limiting likelihood ratio process is discontinuous, a setting outside the scope of classical approaches based on the uniform or Skorokhod $J\_1$ topologies. We derive sufficient conditions for the weak convergence of likelihood ratio processes in the space of c{à}dl{à}g functions on $\RR$ vanishing at $\pm\infty$, endowed with the Skorokhod $M\_1$ topology, and show how this convergence yields the asymptotic behavior of the maximum likelihood estimator. We also introduce new techniques for controlling the $M\_1$ modulus of continuity. Although these techniques are particularly well suited to models in which all jumps and steep continuous transitions are in the same direction, they are of independent interest and may prove useful beyond this setting. As an application, we study a smooth change-point model for inhomogeneous Poisson processes in the fast regime, where the transition interval shrinks faster than $1/n$. We establish that in this case the maximum likelihood estimator has the same asymptotic behavior (consistency, rate of convergence, limiting distribution and convergence of moments) as in the corresponding ''pure'' change-point model.
