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

Abstract

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 \(\theta _{M}\) (or \(\theta _{J}\)) under squared log error loss function. The uniformly minimum risk unbiased (UMRU) estimator of \(\theta _{M}\) and \(\theta _{J}\) are obtained. In the case of \(k=2,\) we give a sufficient condition for minimaxity of an estimator of \(\theta _{M}\) and \(\theta _{J},\) and show that the UMRU and natural estimators of \(\theta _{J}\) are minimax. Also the class of linear admissible estimators of \(\theta _{M}\) and \(\theta _{J}\) 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 \(\theta _{J},\) and show that the UMRU estimator of \(\theta _{J}\) is inadmissible. Finally, we compare the risk of the obtained estimators numerically, and discuss the results for selected uniform population.