To account for overdispersion in count data, that is variation in excess of that justified from the assumed model, one may consider an additional source of variation, by assuming that each observation, <i>Y<sub>i</sub>, i = </i>1, ..., <i>m</i>, arises from a conditionally independent Poisson distribution, given its respective mean <i>q<sub>i</sub>, i = </i>1, ..., <i>m</i>. We review various frequentist methods for the estimation of the Poisson parameters <i>q<sub>i</sub>, i = </i>1, ..., <i>m</i>, which are based on the inadmissibility of the usual unbiased maximum likelihood estimator, in terms of the associated risk in dimensions greater than two. The so called shrinkage estimators adjust the maximum likelihood estimates towards a fixed or data-determined point, abandoning unbiasedness in favour of lower risk. Inferences for the parameters of interest can also be drawn employing Bayesian methods. Conjugate models are often adopted to facilitate the computational procedure. In this thesis we assume a nonconjugate log-normal prior distribution, which allows for more dispersion in the Poisson means and can also accommodate a correlation structure. We derive two empirical Bayes estimators, which approximate the posterior mean. The first is based on a linear shrinkage rule, while the second employs a non-iterative importance sampling technique. The frequency properties of the two estimators in terms of average risk are assessed and compared to other estimating approaches proposed in the literature. A full hierarchical Bayes analysis is also considered, assuming both informative and non-informative prior distributions at the lower stage of the hierarchy. Some analytical posterior inferences, based on simple approximations are obtained. We then employ stochastic simulation techniques, suggesting two Markov chain Monte Carlo methods which involve the Gibbs sampler and a hybrid strategy. They rely on a log-normal/gamma mixture approximation to the full conditional posterior distribution of the parameters <i>q<sub>i</sub></i>, <i>i </i>= 1, ,..., <i>m</i>. The shrinkage behaviour of the hierarchical Bayes estimator is explored, and its average risk is examined through frequency simulations. Examples and applications of the considered methods are given throughout the thesis.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:662556 |
| Date | January 2000 |
| Creators | Streftaris, George |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/14505 |
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