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

5.5 Bootstrap approach to standard error approximation
The standard error of the parameter estimates in a mixture model can be obtained by
approximating the covariance matrix of Φˆ using the inverse of the information matrix.
It is important to mention that these estimates of the covariance matrix of the maximum
likelihood estimation based on the expected or observed information matrices are
guaranteed to be valid inferentially only asymptotically (McLachlan and Peel, 2000). In
particular, for mixture models, it is recognized that the sample size n has to be very
large before the asymptotic theory of maximum likelihood applies (McLachlan and
Peel, 2000). Since our sample size is not too large, we can use a resampling approach to
this problem, the bootstrap method. Basford et al., (1997) and Peel (1998) compared the
bootstrap and information-based approaches for some normal mixture models. They
found that unless the sample size was very large, the standard errors found by the
information-based approach were too unstable to be recommended. In such situations
the bootstrap approach is recommended and we use this approach in this thesis.
The bootstrap approach of calculating the standard error is explained by McLachlan and
Peel (2000). Tthe bootstrap method was first introduced by Efron (1979). Thereafter the
series of articles and books by Efron (1982), Efron and Tibshirani (1993), Davison and
Hinkley (1997), Chernick (1999) were published. Over the past twenty-five years, the
bootstrap method has become one of the most admired developments in statistics.
98