Admissible and minimax estimation of the parameter of the selected Pareto population under squared log error loss function

The problem of estimation after selection arises when we select a population from the given k populations by a selection rule, and estimate the parameter of the selected population. In this paper we consider the problem of estimation of the scale parameter of the selected Pareto population (or ) under squared log error loss function. The uniformly minimum risk unbiased (UMRU) estimator of and are obtained. In the case of we give a sufficient condition for minimaxity of an estimator of and and show that the UMRU and natural estimators of are minimax. Also the class of linear admissible estimators of and are obtained which contain the natural estimator. By using the Brewester-Ziedeck technique we find sufficient condition for inadmissibility of some scale and permutation invariant estimators of and show that the UMRU estimator of is inadmissible. Finally, we compare the risk of the obtained estimators numerically, and discuss the results for selected uniform population.