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Two stage conditionally unbiased estimators of the selected mean

Author

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  • Cohen, Arthur
  • Sackrowitz, Harold B.

Abstract

The problem is to estimate the mean of the selected population. The selection rule is to choose the population with the largest sample mean when such sample means are calculated from the first stage sample. An estimator of the selected mean is unbiased if its expected value equals the expected value of the selected mean. We seek conditionally unbiased estimators of the selected mean given the ordering of the set of sample means based on the first stage sample. Conditionally unbiased estimators are of course unconditionally unbiased. For several distributions such as the normal, with unknown mean, and binomial, no conditionally unbiased estimators exist based on a one stage sample. We propose a two stage sample where observations at stage two are taken from the selected population only. Such a procedure has the advantage of yielding conditionally unbiased estimators and enables, possibly a better allocation of available sample points. We find the uniformly minimum variance conditionally unbiased estimators (UMVCUE) for the normal case when the variance is known or when a common unknown variance is present. We also find the UMVCUE for the gamma case and indicate that the method is suitable for many other cases as well.

Suggested Citation

  • Cohen, Arthur & Sackrowitz, Harold B., 1989. "Two stage conditionally unbiased estimators of the selected mean," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 273-278, August.
    • Handle: RePEc:eee:stapro:v:8:y:1989:i:3:p:273-278
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    Citations

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    Cited by:

    1. Sill, Michael W. & Sampson, Allan R., 2009. "Drop-the-losers design: Binomial case," Computational Statistics & Data Analysis, Elsevier, vol. 53(3), pages 586-595, January.
    2. Deshpande, Jayant V. & Muhammad Fareed, T. P., 1995. "A note on conditionally unbiased estimation after selection," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 17-23, January.
    3. Vellaisamy, P. & Jain, Sushmita, 2008. "Estimating the parameter of the population selected from discrete exponential family," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1076-1087, July.
    4. Kumar, Somesh & Kar, Aditi, 2001. "Estimating quantiles of a selected exponential population," Statistics & Probability Letters, Elsevier, vol. 52(1), pages 9-19, March.
    5. Per Broberg & Frank Miller, 2017. "Conditional estimation in two-stage adaptive designs," Biometrics, The International Biometric Society, vol. 73(3), pages 895-904, September.
    6. Rosenkranz, Gerd K., 2014. "Bootstrap corrections of treatment effect estimates following selection," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 220-227.
    7. P. Vellaisamy & Abraham Punnen, 2002. "Improved estimators for the selected location parameters," Statistical Papers, Springer, vol. 43(2), pages 291-299, April.
    8. Kumar, Somesh & Mahapatra, Ajaya Kumar & Vellaisamy, P., 2009. "Reliability estimation of the selected exponential populations," Statistics & Probability Letters, Elsevier, vol. 79(11), pages 1372-1377, June.
    9. Nigel Stallard & Peter K Kimani, 2018. "Uniformly minimum variance conditionally unbiased estimation in multi-arm multi-stage clinical trials," Biometrika, Biometrika Trust, vol. 105(2), pages 495-501.
    10. Haibing Zhao & Xinping Cui, 2020. "Constructing confidence intervals for selected parameters," Biometrics, The International Biometric Society, vol. 76(4), pages 1098-1108, December.

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