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Journal of Statistical Theory and Practice

, Volume 9, Issue 2, pp 395–418 | Cite as

Estimation After Selection From Gamma Populations with Unequal Known Shape Parameters

  • Mohd. Arshad
  • Neeraj Misra
  • P. Vellaisamy
Article

Abstract

Let π1,…,πk be k (≥ 2) independent gamma populations, where the population πi has an unknown scale parameter θi > 0 and known shape parameter νi > 0, i = 1,…, k. We call the population associated with μ[k] = max {μ1,…, μk}, μi = νiθi the best population. For the goal of selecting the best population, Misra and Arshad (2014) proposed a class \({\mathcal D}_{0}\) of selection rules for the case of (possibly) unequal shape parameters. In this article, we consider the problem of estimating the mean μS of the population selected by a fixed selection rule \({\underline \delta^{\underline a }} \in {\mathcal D}_{0}\), under a scale-invariant loss function. We derive the uniformly minimum variance unbiased estimator (UMVUE). Two other natural estimators φN,1 and φN,2, which are respectively the analogs of the UMVUE and the best scale invariant estimators of μi’s for the component problem, are studied. We show that φN,2 is generalized Bayes with respect to a noninformative prior distribution, and is also minimax when k = 2. The UMVUE and the natural estimator φN,1 are shown to be inadmissible, and better estimators are obtained. A numerical study on the performance of various estimators indicates that the natural estimator φN,2 outperforms the other natural estimators.

Keywords

Gamma populations Inadmissible estimators Scaled-squared error loss Minimax estimator Unequal shape parameters UMVUE 

AMS Subject Classification: 2010: Primary

62F07 

AMS Subject Classification: 2010: Secondary

62F10 62C20 

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Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchAligarh Muslim UniversityAligarhIndia
  2. 2.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKanpurIndia
  3. 3.Department of MathematicsIndian Institute of Technology BombayBombayIndia

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