Empirical Bayes improvement of Kalman filter type of estimators

We consider the problem of estimating the means μi of n random variables Yi ∼ N(μi, 1), i = 1, . . . , n. Assuming some structure on the μ process, e.g., a state space model, one may use a summary statistics for the contribution of the rest of the observations to the estimation of μi. The most important example for this is the Kalman filter. We introduce a nonlinear improvement of the standard weighted average of the given summary statistics and Yi itself, using empirical Bayes methods. The improvement is obtained under mild assumptions. It is strict when the process that governs the states μ1, . . . , μn is not a linear Gaussian state-space model. We consider both the sequential and the retrospective estimation problems.