Estimating After Selection in the Discrete Exponential Family under k−Normalized Squared Error Loss Function

Let p(p ≥ 2) be independent discrete exponential family populations with unknown parameter θi,1 ≤ i ≤ p. Suppose from each population we have an observation Xi. The population corresponding to the largest θi, is the best population. To select a population out of these populations, we consider the natural selection rule which selects the population having the largest Xi and randomly selects in case of ties. Our aim in the paper is to estimate the parameter of the selected population under k−normalized squared error loss function for the important special cases k =0 ,1, and 2. The natural estimator is shown to be risk-biased using Lehmann’s (1951) risk−unbiased concept. For the average worth of the parameters of tied populations with the largest Xi, the UMV U E and UMRUE are also derived. Some applications are presented for the selected Poisson, and negative binomial populations. Mathematics Subject Classification: Primary 62F10; secondary 62F07.