Abstract

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.

Citations

6232	Maximum likelihood from incomplete data via the em algorithm - Dempster, Laird, et al. - 1977
1447	A New Approach to Linear Filtering and Prediction Problems - Kalman, Emil - 1960
1236	Time Series Analysis - Hamilton - 1994
752	Novel approach to nonlinear/non-gaussian bayesian state estimation - Gordon, Salmond, et al. - 1993
653	A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle - Hamilton - 1989
589	AFM. Sampling-based approaches to calculating marginal densities - AE, Smith - 1990
576	Applied Optimal Estimation - Gelb - 1974
555	Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation - Engle - 1982
514	Stochastic processes and filtering theory - Jazwinski - 1970
401	Wong W: The calculation of posterior distributions by data augmentation - Tanner - 1987
295	Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods - Smith, Roberts - 1993
276	Monte Carlo filter and smoother for non-Gaussian nonlinear state space models - Kitagawa - 1996
249	Bayesian Inference in Econometric Models Using Monte Carlo Integration, Econometrica 57 (6 - Geweke - 1989
240	On Gibbs Sampling for State Space Models - Carter, Kohn - 1994
229	Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence - Hall
209	New Results in Linear Filtering and Prediction Theory,” Trans - Kalman, Bucy - 1961
160	An Approach to Time Series Smoothing and Forecasting Using the EM - Shumway, Stoffer - 1982
148	Nonlinear Bayesian estimation using Gaussian sum approximations - Alspach, Sorenson - 1972
121	Non-Gaussian state-space modeling of nonstationary time series - Kitagawa - 1987
112	Bayesian statistics without tears: a samplingresampling perspective. volume 46 - Smith, Gelfand - 1992
102	The Statistical Theory of Linear Systems - Hannan, Deistler - 1988
98	A Monte Carlo approach to nonnormal and nonlinear state-space modeling - Carlin, Polson, et al. - 1992
59	State space modeling of time series - Aoki - 1987
47	Monte Carlo simulation and numerical integration - Geweke - 1996
39	Modelling stochastic volatility: a review and comparative study - Taylor - 1994
32	A self-organising state-space model - Kitagawa - 1998
30	Antihetic acceleration of Monte Carlo integration in Bayesian inference - Geweke - 1988
29	Inference for nonconjugate Bayesian models using the gibbs sampler. The Canadian - Carlin, Polson - 1991
29	Monte Carlo approximations for general state space models - Hürzeler, Künsch - 1998
25	Smoothness Priors Analysis of Time - Kitagawa, Gersch - 1996
19	Unit Roots in Economic Time Series: A Selective Survey - Diebold, Nerlove - 1990
19	A Quasi-Bayesian Approach to Estimating Parameters for Mixtures of Normal Distributions - Hamilton - 1991
18	Robustification of Kalman filter models - Meinhold, Singpurwalla - 1989
16	Estimation, Inference and Forecasting of Time Series Subject to Changes - Hamilton - 1993
13	Nonlinear Filters: Estimation and Applications (Second, Revised and Enlarged Edition - Tanizaki - 1996
11	Data Revision, Reconstruction, and Prediction: An Application to Inventory Investment - Howrey - 1984
11	Prediction of Final Data with Use of Preliminary and/or Revised Data - Mariano, Tanizaki - 1995
10	Estimation in the presence of stochastic parameter variation. Econometrica - Cooley, Prescott - 1976
9	The Kalman Filter: Applications to Forecasting and Rational Expectations Models - Engle, Watson - 1987
9	Prediction, filtering and smoothing in non-linear and non-normal cases using Monte Carlo integration - Tanizaki, Mariano - 1994
8	Bayesian estimation of state-space models using the metropolis-hastings algorithm within gibbs sampling - Geweke, Tanizaki - 2001
7	Kalman filtering estimation of unobserved rational expectations with an application to the German hyperinflation - Burmeister, Wall - 1982
7	Varying coefficient regression - Nicholls, Pagan - 1985
7	Kendall’s Advanced Theory - O’Hagan - 1994
6	Time Series: Theory and Models - Brockwell, Davis - 1987
6	Prediction theory for autoregressive moving average processes - Burridge, Wallis - 1988
6	Time-Varying Regression Coefficients: A Mixed Estimation Approach and - Cooper
6	On Markov Chain Monte Carlo Methods for Nonlinear and Non-Gaussian State-Space Models - Geweke, Tanizaki - 1999
6	Nonlinear and non-Gaussian state space modeling using sampling techniques - Tanizaki - 2001
5	The Art of Computer Generation of Random Variables - Boswell, Gore, et al. - 1993