Springer - Read

298 Chapter 8 State-Space Models
Figure 8-6
Monthly number of
U.S. cases of polio,
Jan. ’70–Dec. ’83.
density is assumed to be Poisson with mean exp{xt}, i.e.,
p(yt|xt) ext yt −e e xt
, yt0, 1,..., (8.8.21)
yt!
while the state variables are assumed to follow a regression model with Gaussian
AR(1) noise. If ut (ut1,...,utk) ′ are the regression variables, then
Xt β ′ ut + Wt, (8.8.22)
where β is a k-dimensional regression parameter and
Wt φWt−1 + Zt, {Zt} ∼IID N 0,σ 2 .
The transition density function for the state variables is then
p(xt+1|xt) n(xt+1; β ′ ut+1 + φ xt − β ′ ut), σ 2 . (8.8.23)
The case σ 2 0 corresponds to a log-linear model with Poisson noise.
Estimation of the parameters θ β ′ ,φ,σ2 ′
in the model by direct numerical
maximization of the likelihood function is difficult, since the likelihood cannot be
written down in closed form. (From (8.8.3) the likelihood is the n-fold integral,
∞
∞ n
··· exp xtyt − e xt
L θ; x (n) n
(dx1 ···dxn) (yi!),
−∞
−∞
t1
where L(θ; x) is the likelihood based on X1,...,Xn.) To overcome this difficulty,
Chan and Ledolter (1995) proposed an algorithm, called Monte Carlo EM (MCEM),
whose iterates θ (i) converge to the maximum likelihood estimate. To apply this algorithm,
first note that the conditional distribution of Y (n) given X (n) does not depend
0 2 4 6 8 10 12 14
o o
ooo
o
oo
o oo
o
o
o
o o
o
o
o
o
oo
o oo
o o ooo oooo
ooo
oo
o o oo oo o oo
oo
oo
o o o oo oooo
ooooo
oo
o oo o ooo
o o oo oo
o
o
o
o o o o
o
o
o o
o
o
oo o o o o
oo
o
o
o o
o
o
oo
o oo o o o
o o
o
ooo
o o ooo
ooo o ooo o oo ooooo
ooo
o
1970 1972 1974 1976 1978 1980 1982 1984
o
o
i1