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

Author

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  • Lakshmi Kanta Patra

    (Indian Institute of Petroleum and Energy)

  • Suchandan Kayal

    (National Institute of Technology Rourkela)

  • Somesh Kumar

    (Indian Institute of Technology Kharagpur)

Abstract

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.

Suggested Citation

  • Lakshmi Kanta Patra & Suchandan Kayal & Somesh Kumar, 2021. "Minimax estimation of the common variance and precision of two normal populations with ordered restricted means," Statistical Papers, Springer, vol. 62(1), pages 209-233, February.
    • Handle: RePEc:spr:stpapr:v:62:y:2021:i:1:d:10.1007_s00362-019-01090-2
    DOI: 10.1007/s00362-019-01090-2
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    References listed on IDEAS

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    1. Constantinos Petropoulos, 2017. "Estimation of the order restricted scale parameters for two populations from the Lomax distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(4), pages 483-502, May.
    2. Manas Tripathy & Somesh Kumar & Nabendu Pal, 2013. "Estimating common standard deviation of two normal populations with ordered means," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 22(3), pages 305-318, August.
    3. Mohd. Arshad & Neeraj Misra, 2016. "Estimation after selection from exponential populations with unequal scale parameters," Statistical Papers, Springer, vol. 57(3), pages 605-621, September.
    4. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2008. "Stein's phenomenon in estimation of means restricted to a polyhedral convex cone," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 141-164, January.
    5. Yogesh Mani Tripathi & Constantinos Petropoulos & Mayank Jha, 2018. "Estimation of the shape parameter of a Pareto distribution," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(18), pages 4459-4468, September.
    Full references (including those not matched with items on IDEAS)

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