Since Kitagawa (1987) and Kramer and Sorenson (1988) proposed the filter and smoother using numerical integration, nonlinear and/or non-Gaussian state estimation problems have been developed. Numerical integration becomes extremely computer-intensive in the higher dimensional cases of the state vector. Therefore, to improve the above problem, the sampling techniques such as Monte Carlo integration with importance sampling, resampling, rejection sampling, Markov chain Monte Carlo and so on are utilized, which can be easily applied to multi-dimensional cases. Thus, in the last decade, several kinds of nonlinear and non-Gaussian filters and smoothers have been proposed using various computational techniques. The objective of this paper is to introduce the nonlinear and non-Gaussian filters and smoothers which can be applied to any nonlinear and/or non-Gaussian cases. Moreover, by Monte Carlo studies, each procedure is compared by the root mean square error criterion.

In this paper, a nonlinear and/or non-Gaussian smoother utilizing Markov chain Monte Carlo Methods is proposed, where the measurement and transition equations are specified in any general formulation and the error terms in the state-space model are not necessarily normal. The random draws are directly generated from the smoothing densities. For random number generation, the Metropolis-Hastings algorithm and the Gibbs sampling technique are utilized. The proposed procedure is very simple and easy for programming, compared with the existing nonlinear and non-Gaussian smoothing techniques. Moreover, taking several candidates of the proposal density function, we examine precision of the proposed estimator.

The rejection sampling filter and smoother, proposed by Tanizaki (1996, 1999), Tanizaki and Mariano (1998) and Hurzeler and Kunsch (1998), take a lot of time computationally. The Markov chain Monte Carlo smoother, developed by Carlin, Polson and Sto#er (1992), Carter and Kohn (1994, 1996) and Geweke and Tanizaki (1999a, 1999b), does not show a good performance depending on nonlinearity and nonnormality of the system in the sense of the root mean square error criterion, which reason comes from slow convergence of the Gibbs sampler. Taking into account these problems, we propose the nonlinear and non-Gaussian filter and smoother which have much less computational burden and give us relatively better state estimates, although the proposed estimator does not yield the optimal state estimates in the sense of the minimum mean square error. The proposed filter and smoother are called the quasi-optimal filter and quasi-optimal smoother in this paper. Finally, through some Monte Carlo studies, the quasi-optimal filter and smoother are compared with the rejection sampling procedure and the Markov chain Monte Carlo procedure. 1

: For the last decade, various simulation-based nonlinear

Fiducial pdf as special case in either Bayesian or frequentist approach; equivalence to the information-metric “prior”; relevance to parameter estimation.