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