Estimation of Poisson parameters: maximum likelihood, Bayes, empirical Bayes or a compromise?
This thesis deals with three point estimation problems where the random variables are assumed to be Poisson with parameters following conjugate prior distributions. In all the three problems, we try to estimate a p-variate Poisson mean;For the first problem, a class of estimators compromising between the maximum likelihood and the Bayes estimators is proposed; the proposed estimators are named limiting translation rule (LTR). In the second problem, we propose the so-called limiting translation compound Bayes rules (LTCBR) which compromise between the maximum likelihood and empirical Bayes estimators. Finally, in the third problem under the assumption that the p-variate mean can be divided into two natural groups, we compare the combined against the separate estimators;We show that the LTRs perform satisfactorily through both the risk criterion and the Bayes risk criterion, the latter measured by what is defined as relative saving loss. With regard to the LTCBRs, it is shown that these estimators have both good componentwise risk and Bayes risk performance, the latter again measured by the relative saving loss. The study of the combined against the separate estimators shows the preference of combined estimators when the scales parameters of the prior distributions assigned to each group are close to each other, otherwise the use of the separate estimators seems more appropriate.