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

The bootstrap method is a powerful and important technique permits the variability in a
random quantity to be assessed using just the data at hand (McLachlan and Peel, 2000).
An estimate Fˆ of the underlying distribution is formed from the observed data Y .
Conditional on the latter, the sampling distribution of the random quantity of interest
with F replaced by Fˆ defines its so-called bootstrap distribution, which provides an
approximation to its true distribution. It is assumed that Fˆ has been so formed that the
stochastic structure of the model has been preserved. Usually, it is impossible to express
the bootstrap distribution in a simple form, and it must be approximated by Monte Carlo
methods whereby pseudo-random samples (bootstrap samples) are drawn from Fˆ . If a
parametric form is adopted for the distribution function of Y , where Φ denotes the
vector of unknown parameters, and then the parametric bootstrap uses an estimate Φˆ
formed from y in place of Φ. That is, if we write F as F
Φ
to signify its dependence
on Φ, then the bootstrap data are generated from
F ˆ F .
=
Φ ˆ
McLachlan and Peel (2000) explained that the standard error estimation of Φˆ could be
stated using the bootstrap method by the following steps:
Step 1: The new data,
*
Y , called the bootstrap sample, is generated according to Fˆ , an
estimate of the distribution formed from the original observed data Y. That is, in the
case where Y contains the observed values of a random sample of size n,
the observed values of the random sample
* * *
Y
1, Y
2,..., Yn
~ ,
iid .. .
F
*
Y consists of
99