Estimating parameters of a selected Pareto population

Let Π1,…,Πk be k populations with Πi being Pareto with unknown scale parameter αi and known shape parameter βi;i=1,…,k. Suppose independent random samples (Xi1,…,Xin), i=1,…,k of equal size are drawn from each of k populations and let Xi denote the smallest observation of the ith sample. The population corresponding to the largest Xi is selected. We consider the problem of estimating the scale parameter of the selected population and obtain the uniformly minimum variance unbiased estimator (UMVUE) when the shape parameters are assumed to be equal. An admissible class of linear estimators is derived. Further, a general inadmissibility result for the scale equivariant estimators is proved.