By Tanizaki H.
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Extra info for Nonlinear and Non-Gaussian State-Space Modeling with Monte Carlo Techniques: A Survey and Comparative Study
T and ηt are assumed to be mutually independent. We compare the extended Kalman ﬁlter and smoother16 and the nonlinear and nonGaussian ﬁlters and smoothers introduced in Section 3. The simulation procedure is as follows: (i) generating random numbers of t and ηt for t = 1, 2, · · · , T , we obtain a set of data yt and αt , t = 1, 2, · · · , T , from equations (1) and (2), where T = 20, 40, 100 is taken, (ii) given YT , perform each estimator, (iii) repeat (i) and (ii) G times and compare the root mean 1/2 square error (RMSE) for each estimator.
Let P∗ (z|x) be the proposal density, which is the conditional distribution of z given x. We should choose the proposal density P∗ (z|x) such that random draws can be easily and quickly generated. Deﬁne the acceptance probability ω(x, z) as follows: min ω(x, z) = 1, P (z|Ai,t−1 , A∗i−1,t+1 , YT )P∗ (x|z) ,1 , P (x|Ai,t−1 , A∗i−1,t+1 , YT )P∗ (z|x) if P (x|Ai,t−1 , A∗i−1,t+1 , YT )P∗ (z|x) > 0, otherwise. To generate random draws from P (AT |YT ), the following procedure is taken: (i) pick up appropriate values for α1,0 and α0,t , t = 1, 2, · · · , T , (ii) generate a random draw z from P∗ (·|αi−1,t ) and a uniform random draw u from the uniform distribution between zero and one, (iii) set αi,t = z if u ≤ ω(αi−1,t , z) and set αi,t = αi−1,t otherwise, (iv) repeat (ii) and (iii) for t = 1, 2, · · · , T , and (v) repeat (ii) – (iv) for i = 1, 2, · · · , N .
However, the disadvantages of rejection sampling are: (i) we need to compute a, which sometimes does not exist and (ii) it takes a long time when the acceptance probability ω(·) is close to zero. See, for example, Knuth (1981), Boswell, Gore, Patil and Taillie (1993), O’Hagan (1994) and Geweke (1996) for rejection sampling. Gibbs Sampling: Geman and Geman (1984), Tanner and Wong (1987), Gelfand and Smith (1990), Gelfand, Hills, Racine-Poon and Smith (1990) and so on developed the Gibbs sampling theory, which is concisely described as follows (also see Geweke (1996, 1997)).