This work is devoted to simultaneously estimating the parameters of the distributions of several independent Poisson random variables. In particular, we explore the possibility of finding estimators of the Poisson parameters which have better performance than the maximum likelihood estimator (MLE). We first approach the problem from a fre-quentist point of view, employing a generally scaled loss function, called the k-normalized squared error loss function
L[sub k] (λ, [sup ^]λ) = [sup P] ∑ [sub i=1] (λ[sub i] – [sup ^]λ[sub i])²/ λ[sup k][sub i],
where k is a non-negative integer. The case k=0' is the squared error loss case, in which we propose a large class of estimators including those proposed by Peng  as special cases. Estimators pulling the MLE towards a point other than zero as well as a point determined by the data itself are proposed, and it is shown that these estimators dominate the MLE uniformly. Under L[sub k] with k ≥ 1, we obtain a class of estimators dominating the MLE which includes the estimators proposed by Clevenson and Zidek .
We next approach the problem from a Bayesian point, of view; a two-stage prior distribution is adopted and results for a large class of prior distributions are derived. Substantial savings in terms of mean squared error loss of the Bayes point estimators over the MLE are expected, especially when the Poisson parameters fall into a relatively narrow range.
An empirical Bayes approach to the problem is carried out along the line suggested by Clevenson and Zidek . Some results are obtained which parallel those of Efron and Morris , who work under the assumption that the random variables are normally distributed.
We report the results of our computer simulation to quantitatively examine the performance of some of our proposed estimators. In most cases, the savings, under the appropriate loss functions, are an increasing function of the number of Poisson parameters. The simulation results indicate that our estimators are very promising. The savings of the Bayes estimators depend on the choice of prior hyperparameters, and hence proper choice leads to substantial improvement over the MLE.
Although most of the results in this work are derived under the assumption that only one observation is taken from each Poisson distribution,
we extend some results to the case where possibly more than one observation is taken. We conclude with suggestions for further work. / Science, Faculty of / Mathematics, Department of / Graduate
|University of British Columbia
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