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

Nader Nematollahi ()
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Nader Nematollahi: Allameh Tabataba’i University

Statistical Papers, 2017, vol. 58, issue 2, No 3, 319-339

Abstract: 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 $$\theta _{M}$$ 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.

Keywords: Admissibility; Minimaxity; Pareto distribution; Selected population; Squared log error loss function; Uniformly minimum risk unbiased estimator; Primary 62F07; Secondary 62F10; 62C15; 62C20 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00362-015-0699-6

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