This paper revisits classical work of Rauch et al. (1965) and develops a novel statistical method for maximum likelihood (ML) recursive state estimation in general state–space models. The new method is based on statistical estimation theory for incomplete information, which has been well developed primarily for ML parameter estimation (Dempster et al., 1977). Distributional identities for the posterior score function and information matrix of state are established. Using these identities, a fast convergent EM-gradient algorithm is proposed, extending the Lange (1995) algorithm for ML recursive state estimation. It revisits and provides an improvement to the EM-algorithm of Ramadan and Bitmead (2022). An explicit form of the information matrix is developed to provide empirical estimates of the standard errors. Sequential Monte Carlo method is used for the valuation of the score function, information and posterior covariance matrices. Some numerical examples are discussed to exemplify the main results.

中文翻译：

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最大似然递归状态估计：一种基于不完全信息的方法


本文回顾了 Rauch 等人的经典著作。 （1965）并开发了一种新的统计方法，用于一般状态空间模型中的最大似然（ML）递归状态估计。新方法基于不完全信息的统计估计理论，该理论主要用于 ML 参数估计（Dempster 等，1977）。建立了后验得分函数和状态信息矩阵的分布恒等式。利用这些恒等式，提出了一种快速收敛 EM 梯度算法，扩展了用于 ML 递归状态估计的 Lange (1995) 算法。它重新审视并改进了 Ramadan 和 Bitmead (2022) 的 EM 算法。开发了信息矩阵的显式形式来提供标准误差的经验估计。序贯蒙特卡罗方法用于评估得分函数、信息和后验协方差矩阵。讨论了一些数值例子来举例说明主要结果。