Nonparametric interpolation of the Markov sequence

Consideration was given to the problem of interpolation (smoothing) of the nonobservable component of the composite Markov process within the framework of the conditional Markov scheme. In the case of the dynamic observation models such as autoregression, equations were derived for the a posteriori interpolation density of the probability of the state of the nonobservable component. The aim of the present paper was to construct a smoothing algorithm for an unknown family of the distributions of the nonobservable component of the partially observable random Markov sequence. The result was obtained for the strictly stationary random Markov processes with mixing and for the conditional densities in the observation model from the exponential family of distributions. Computer-aided modeling within the framework of the Kalman scheme demonstrated that the sampled root-mean-square error of the nonparametric smoothing algorithm constructed for an unknown state equation was situated between the errors of the optimal linear filtration and the optimal linear interpolation.