Minimax estimation of the common variance and precision of two normal populations with ordered restricted means

  • Lakshmi Kanta PatraEmail author
  • Suchandan Kayal
  • Somesh Kumar
Regular Article


Consider two independent normal populations with a common variance and ordered means. For this model, we study the problem of estimating a common variance and a common precision with respect to a general class of scale invariant loss functions. A general minimaxity result is established for estimating the common variance. It is shown that the best affine equivariant estimator and the restricted maximum likelihood estimator are inadmissible. In this direction, we derive a Stein-type improved estimator. We further derive a smooth estimator which improves upon the best affine equivariant estimator. In particular, various scale invariant loss functions are considered and several improved estimators are presented. Furthermore, a simulation study is performed to find the performance of the improved estimators developed in this paper. Similar results are obtained for the problem of estimating a common precision for the stated model under a general class of scale invariant loss functions.


Restricted maximum likelihood estimator Scale invariant loss function Minimaxity Stein-type estimator Brewster and Zidek-type estimator 

Mathematics Subject Classification

62F10 62C20 



The authors would like to thank two anonymous reviewers for their valuable comments, which have improved the presentation of this paper. The authors also thank to Prof. Éric Marchand for his valuable suggestions which have helped to improve the manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Lakshmi Kanta Patra
    • 1
    Email author
  • Suchandan Kayal
    • 2
  • Somesh Kumar
    • 3
  1. 1.Indian Institute of Petroleum and EnergyVisakhapatnamIndia
  2. 2.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia
  3. 3.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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