The rejection sampling filter and smoother, proposed by Tanizaki (1996, 1999), Tanizaki and Mariano (1998) and Hürzeler and Künsch (1998), take a lot of time computationally. The Markov chain Monte Carlo smoother, developed by Carlin, Polson and Stoffer (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.

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