Simultaneous Estimation of the Parameters of Independent Poisson Distributions

Abstract

Let be independent Poisson random variables, [Image65.gif] having a Poisson distribution with [Image66.gif]. We want to estimate the unknown parameters [Image67.gif] under squared error loss [Image68.gif]. A formula which Hudson has used in his dissertation [1974] is given for an unbiased estimate of the parameter [Image69.gif]. This formula is used to prove that the estimator [Image70.gif] dominates the usual maximum likelihood estimator [Image71.gif] uniformly if [Image72.gif], where [Image73.gif] is the number of zero observations, [Image74.gif] is the random vector with coordinates [Image75.gif], [Image76.gif], and [Image77.gif]. The estimator [Image78.gif] is proved to be admissible if [Image79.gif]. This is a direct consequence of a more general theorem concerning the admissibility of the M. L. E. for estimating the mean (or canonical parameter) [Image80.gif] of some exponential families [Image81.gif]. The proposed estimator [Image82.gif] is further refined so as to have the desirable property of not always estimating [Image83.gif] by 0 if [Image84.gif], while still dominating the M. L. E. uniformly. Some remarks are made on directions in which this work ought to be extended.