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Point and Interval Estimation of Powers of Scale Parameters for Two Normal Populations with a Common Mean

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  • Pravash Jena

    (National Institute of Technology Rourkela)

  • Manas Ranjan Tripathy

    (National Institute of Technology Rourkela)

  • Somesh Kumar

    (Indian Institute of Technology Kharagpur)

Abstract

The problems of point and interval estimation of powers of scale parameter $$(\sigma _{1}^{2c})$$ ( σ 1 2 c ) have been considered when samples are available from two normal populations with a common mean. Maximum likelihood estimators (MLEs) and plug-in estimators using some of the popular estimators of the common mean have been proposed. A sufficient condition for improving affine equivariant estimators using the quadratic loss function is derived. Moreover, we propose several interval estimators, such as the asymptotic confidence interval, bootstrap confidence intervals, HPD credible interval, and intervals based on generalized pivot variables. Interestingly, some of the well-known estimators for the common mean have been used in constructing the generalized confidence intervals. A numerical comparison among all the proposed estimators has been made in terms of risk (in the case of point estimation) values using the quadratic loss function and coverage probabilities, and average lengths (in the case of interval estimation). Based on our simulation results, some recommendations are given for the use of the estimators. A real-life example has been considered to demonstrate the estimation methods.

Suggested Citation

  • Pravash Jena & Manas Ranjan Tripathy & Somesh Kumar, 2023. "Point and Interval Estimation of Powers of Scale Parameters for Two Normal Populations with a Common Mean," Statistical Papers, Springer, vol. 64(5), pages 1775-1804, October.
  • Handle: RePEc:spr:stpapr:v:64:y:2023:i:5:d:10.1007_s00362-022-01361-5
    DOI: 10.1007/s00362-022-01361-5
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    References listed on IDEAS

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    1. 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.
    2. Pal, Nabendu & Ling, Chiahua, 1995. "Improved minimax estimation of powers of the variance of a multivariate normal distribution under the entropy loss function," Statistics & Probability Letters, Elsevier, vol. 24(3), pages 205-211, August.
    3. Pal, Nabendu & Lin, Jyh-Jiuan & Chang, Ching-Hui & Kumar, Somesh, 2007. "A revisit to the common mean problem: Comparing the maximum likelihood estimator with the Graybill-Deal estimator," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5673-5681, August.
    4. K. Krishnamoorthy & Yong Lu, 2003. "Inferences on the Common Mean of Several Normal Populations Based on the Generalized Variable Method," Biometrics, The International Biometric Society, vol. 59(2), pages 237-247, June.
    5. Nobuo Shinozaki, 1995. "Some modifications of improved estimators of a normal variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(2), pages 273-286, June.
    6. Chang, Ching-Hui & Pal, Nabendu, 2008. "Testing on the common mean of several normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 321-333, December.
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