Simultaneous Estimation of Multiple Poisson Parameters Under Weighted Squared Error Loss.

This paper is concerned with estimation of means of several independent Poisson distributions under weighted squared error loss function. Better estimators in terms of risk are proposed and compared to the usual estimator which is the maximum likelihood and the minimum variance unbiased estimator. Under the loss function, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) estimators which shrink the MLE towards the origin, (delta)('1)(X) = (1-w(Z))X, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) 0 (LESSTHEQ) w(Z) (LESSTHEQ) 1, are considered and conditions on w(Z) are found for (delta)('1)(X) to dominate the usual estimator under the loss function L(,1). It is shown that (delta)('1)(X) also dominates the usual estimator under the various loss functions L(,k), k = 2,3... This generalizes the results of Clevenson and Zidek(1975) and Tsui and Press(1982). Under the more general loss function, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) estimators of the form, (delta)(X) = (1-g(,i)(c)f(,i)(X))X, (delta)(X) = (1-w(,i)(c,X))X, and (delta)(X) = (1-w(c,X))X, where i = 1,..,p, 0 (LESSTHEQ) w(,i)(c,X), w(c,X) (LESSTHEQ) 1, are considered. The relation between the estimators under the loss function L(,k)('c) and the case when more than one observation is taken from some population is studied. We derive an estimator which dominates Tsui and Press's estimator (delta)('TP) when certain conditions on the c(,i)'s hold. We also derive an estimator which dominates the MLE under the various loss function L(,k)('c) for all k (GREATERTHEQ) 1. Estimators which shrink towards other than the origin are developed. We discuss the harmonic mean of the observations as a reasonable center and develop estimators which shrink the MLE towards the harmonic mean under the loss function L(,1). Also, better estimators which incorporate a prechosen point are developed. The risk performance of these derived estimators is compared to that of the usual estimator by simulation. The percentage reduction of average loss(PRAL) of derived estimators compared to the usual estimator(MLE) under the loss function L(,1) is estimated for p = 4, 5 and 8. Computer simulation shows that the risk performance of the derived estimators is remarkably good when all the means are greater than three.PhDStatisticsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/160162/1/8422266.pd