  This paper revisits the work of Rauch et al. (1965) and develops a novel
method for recursive maximum likelihood particle filtering for general
state-space models. The new method is based on statistical analysis of
incomplete observations of the systems. Score function and conditional observed
information of the incomplete observations/data are introduced and their
distributional properties are discussed. Some identities concerning the score
function and information matrices of the incomplete data are derived. Maximum
likelihood estimation of state-vector is presented in terms of the score
function and observed information matrices. In particular, to deal with
nonlinear state-space, a sequential Monte Carlo method is developed. It is
given recursively by an EM-gradient-particle filtering which extends the work
of Lange (1995) for state estimation. To derive covariance matrix of
state-estimation errors, an explicit form of observed information matrix is
proposed. It extends Louis (1982) general formula for the same matrix to
state-vector estimation. Under (Neumann) boundary conditions of state
transition probability distribution, the inverse of this matrix coincides with
the Cramer-Rao lower bound on the covariance matrix of estimation errors of
unbiased state-estimator. In the case of linear models, the method shows that
the Kalman filter is a fully efficient state estimator whose covariance matrix
of estimation error coincides with the Cramer-Rao lower bound. Some numerical
examples are discussed to exemplify the main results.
