Simultaneous Estimation of Poisson Parameters
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This paper is devoted to the simultaneous estimation of the means of p≥ 2 independent Poisson distributions. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson--Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.
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