multivariate poisson hidden markov models for analysis of spatial ...

where the estimate Fˆ (now denoting the distribution function of a single observation
Y ) is held fixed at its observed value.
j
Step 2: The EM algorithm is applied to the bootstrap observed data
*
Y to compute the
maximum likelihood estimates for this dataset,
* ˆΦ .
Step 3: The bootstrap covariance matrix of
* ˆΦ is given by
cov ( ˆ
where
*
[{ ˆ *
( ˆ )}{ ˆ * *
)
( ˆ *
= E Φ − E Φ Φ − E Φ )} ],
(5.33)
* T
Φ *
*
E denotes expectation over the bootstrap distribution specified by Fˆ .
The bootstrap covariance matrix can be approximated by Monte Carlo methods. Steps
(1) and (2) are repeated independently several times (say, B) to give B independent
realizations of
* ˆΦ , denoted by
Φ ˆ
*
1
,...., ˆ
*
Φ B
. Then (5.33) can be approximated by the
sample covariance matrix of these B bootstrap replications to give
B
ˆ * ˆ *
)( ˆ *
(
ˆ * T
∑ Φ − − )
*
cov ( ˆ * 1
) ≈ =
b
Φ Φb
Φ
b
Φ (5.34)
( B −1)
where
B
*
∑Φ
ˆ
b
Φ ˆ * = =1 .
B
The standard error of the i th element of Φ can be estimated by the positive square toot
of the i th diagonal element of (5.34). It has been demonstrated that 50 to 100 bootstrap
100