发表机构
Inria Lyon; Univ. Lille, CNRS, IMT Nord Europe, Inria（法国国家信息与自动化研究所里昂分部; 里尔大学、法国国家科学研究中心、北欧高等理工学院、法国国家信息与自动化研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对马尔可夫链不变分布估计，提出前瞻估计量并证明其在可逆核下渐近优于经验估计量，引入插入式版本并给出优于经验估计量的充要条件，模拟与真实数据验证了效果。
AI 中文摘要
我们考虑在求解不变方程不可行时，对离散状态空间上马尔可夫核的不变分布 $\pi$（或在其下泛函 $f$ 的均值 $\pi(f)$）进行估计的问题。前瞻估计量通过将经验占据测度沿链传播 $k$ 步来利用核的知识。当核已知时（如在马尔可夫链蒙特卡洛中），我们比较其与经验估计量在独立或马尔可夫数据、平稳或非平稳情形下的均方误差。我们的主要结果证明，对于马尔可夫数据，一步估计量在渐近意义上总是优于经验估计量，且对于可逆核，改进随 $k$ 增大而增加。一个不可逆的反例表明可逆性不能简单舍弃。在核属于参数族且参数未知的统计设定中，我们引入插入式前瞻估计量，其使用估计的核执行前瞻步骤。当参数以参数速率估计时，我们给出插入式前瞻估计量优于经验估计量的一个充要条件。理论结果通过广泛的模拟研究加以说明，这些研究也进一步揭示了所考虑估计量的行为。最后，两种设定均应用于法国里昂大都市区地理流动性问题产生的真实数据。
英文摘要
We consider the estimation of the invariant distribution $π$ of a Markov kernel on a discrete state space (or of the mean value $π(f)$ of a functional $f$ under $π$) when solving the invariance equation is not feasible. Look-ahead estimators exploit the knowledge of the kernel by propagating the empirical occupation measure through $k$ steps of the chain. When the kernel is known, as in Markov chain Monte Carlo, we compare their mean squared error with that of the empirical estimator, for independent or Markovian data, stationary or not. Our main result establishes that, for Markovian data, the one-step estimator always improves on the empirical one asymptotically, and that the improvement grows with $k$ for reversible kernels. A non-reversible counterexample shows that reversibility cannot simply be dropped. In the statistical setting where the kernel belongs to a parametric family with unknown parameter, we introduce plug-in look-ahead estimators, which perform the look-ahead step with an estimated kernel. When the parameter is estimated at the parametric rate, we give a necessary and sufficient condition under which the plug-in look-ahead estimator outperforms the empirical one. The theoretical results are illustrated by extensive simulation studies, which also shed further light on the behaviour of the estimators at hand. Both settings are finally applied to real data arising from geographical mobility problems in the Lyon metropolitan area, France.
Comments75 pages
