Abstract
In this paper, the nonlinear non-Gaussian filters and smoothers are proposed using the joint density of the state variables, where the sampling techniques such as rejection sampling (RS), importance resampling (IR) and the Metropolis-Hastings independence sampling (MH) are utilized. Utilizing the random draws generated from the joint density, the density-based recursive algorithms on filtering and smoothing can be obtained. Furthermore, taking into account possibility of structural changes and outliers during the estimation period, the appropriately chosen sampling density is possibly introduced into the suggested nonlinear non-Gaussian filtering and smoothing procedures. Finally, through Monte Carlo simulation studies, the suggested filters and smoothers are examined.
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Bollerslev, T., Engle, R. F. and Nelson, D. B. (1994). ARCH models, Handbook of Econometrics, 4 (eds. R. F. Engle and D. L. McFadden), 2959-3038, Elsevier Science B. V., Amsterdam.
Boswell, M. T., Gore, S. D., Patil, G. P. and Taillie, C. (1993). The art of computer generation of random variables, Handbook of Statist., 9 (ed. C. R. Rao), 661-721, Elsevier Science B.V., Amsterdam.
Carlin, B. P., Polson, N. G. and Stoffer, D. S. (1992). A Monte Carlo approach to nonnormal and nonlinear state space modeling, J. Amer. Statist. Assoc., 87, 493-500.
Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models, Biometrika, 81, 541-553.
Carter, C. K. and Kohn, R. (1996). Markov Chain Monte Carlo in conditionally Gaussian state space models, Biometrika, 83, 589-601.
Chib, S. and Greenberg, E. (1995). Understanding the Metropolis-Hastings algorithm, Amer. Statist., 49, 327-335.
Doucet, A., Godsill, S. and Andrieu, C., (2000). On sequential Monte Carlo sampling methods for Bayesian filtering, Statist. Comput., 10, 197-208.
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of variance of U. K. inflation, Econometrica, 50, 987-1008.
Geweke, J. (1996). Monte Carlo simulation and numerical integration, Handbook of Computational Economics, 1 (eds. H. M. Amman, D. A. Kendrick and J. Rust), 731-800, Elsevier Science B. V., Amsterdam.
Geweke, J. and Tanizaki, H. (1999). On Markov Chain Monte-Carlo methods for nonlinear and non-Gaussian state-space models, Comm. Statist. Simulation Comput., 28, 867-894.
Ghysels, E., Harvey, A. C. and Renault, E. (1996). Stochastic volatility, Handbook of Statist., 14 (eds. G. S. Maddala and C. R. Rao), 119-191, Elsevier Science B. V., Amsterdam.
Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proceedings-F, 140, 107-113.
Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, Cambridge/New York.
Hürzeler, M. and Künsch, H. R. (1998). Monte Carlo approximations for general state-space models, J. Comput. Graph. Statist., 7, 175-193.
Kitagawa, G. (1987). Non-Gaussian state-space modeling of nonstationary time series (with discussion), J. Amer. Statist. Assoc., 82, 1032-1063.
Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state-space models, J. Comput. Graph. Statist., 5, 1-25.
Kitagawa, G. (1998). A self-organizing state-space model, J. Amer. Statist. Assoc., 93, 1203-1215.
Kitagawa, G. and Gersch, W. (1996). Smoothness Priors Analysis of Time Series, Lecture Notes in Statist., No.116, Springer, New York.
Kong, A., Liu, J. S. and Chen, R. (1994). Sequential imputations and Bayesian missing data problems, J. Amer. Statist. Assoc., 89, 278-288.
Liu, J. S. (1996). Metropolized independent sampling with comparisons to rejection sampling and importance sampling, Statist. Comput., 6, 113-119.
Liu, J. S. and Chen, R. (1995). Blind deconvolution via sequential imputations, J. Amer. Statist. Assoc., 90, 567-576.
Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems, J. Amer. Statist. Assoc., 93, 1032-1044.
Liu, J. S., Chen, R. and Wong, W. HG. (1998). Rejection control and sequential importance sampling, J. Amer. Statist. Assoc., 93, 1022-1031.
O'Hagan, A. (1994). Kendall's advanced theory of statistics, 2B, Edward Arnold, London.
Smith, A. F. M. and Gelfand, A. E. (1992). Bayesian statistics without tears: A sampling-resampling perspective, Amer. Statist., 46, 84-88.
Tanizaki, H. (1996). Nonlinear Filters: Estimation and Applications (2nd, revised and enlarged ed.), Springer, Berlin/Heidelberg.
Tanizaki, H. (1999). On the nonlinear and nonnormal filter using rejection sampling, IEEE Trans. Automat. Control, 44, 314-319.
Tanizaki, H. (2001). Nonlinear and non-Gaussian state-space modeling with Monte Carlo techniques: A survey and comparative study, Handbook of Statist (Stochastic Processes: Modeling and Simulation)., (eds. C. R. Rao and D. N. Shanbhag), Elsevier Science B. V., Amsterdam, forthcoming.
Tanizaki, H. and Mariano, R. S. (1996). Nonlinear filters based on Taylor series expansions, Comm. Statist. Theory Methods, 25, 1261-1282.
Tanizaki, H. and Mariano, R. S. (1998). Nonlinear and non-Gaussian state-space modeling with Monte Carlo simulations, J. Econometrics., 83, 263-290.
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Tanizaki, H. Nonlinear and Non-Gaussian State Space Modeling Using Sampling Techniques. Annals of the Institute of Statistical Mathematics 53, 63–81 (2001). https://doi.org/10.1023/A:1017916420893
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DOI: https://doi.org/10.1023/A:1017916420893