Maximum likelihood recursive state estimation: An incomplete-information based approach

This paper revisits classical work of \citet{Rauch} and develops a novel method for maximum likelihood (ML) recursive state estimation in 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 \citep{Dempster}. Score function and posterior information matrix of state are introduced and their distributional identities are derived. Using these identities, a fast convergent EM-gradient-particle algorithm is proposed extending \cite{Lange} algorithm for ML state estimation. Under Neumann boundary condition on posterior density function of state, posterior covariance matrix of estimation error coincides with a Cram\'er-Rao lower bound. In the absence of Neumann boundary condition, the covariance matrix takes a similar form to Huber sandwich estimator \citep{Freedman}. An explicit form of posterior information matrix is developed for calculation of posterior standard errors of ML estimates. Sequential Monte Carlo method is proposed for valuation of the score function, information and posterior covariance matrices. For linear Gaussian state-space model, the method shows that the \cite{Kalman} filter is a fully efficient unbiased ML state estimator. Some numerical examples are discussed to exemplify the main results.