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On a family of distributions attaining the Bhattacharyya bound

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Abstract

A family of distributions for which an unbiased estimator of a functiong(θ) of a real parameter θ can attain the second order Bhattacharyya lower bound is derived. Indeed, we obtain a necessary and sufficient condition for the attainment of the second order Bhattacharyya bound for a family of mixtures of distributions which belong to the exponential family. Furthermore, we give an example which does not satisfy this condition, but where the Bhattacharyya bound is attainable for a non-exponential family of distributions.

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References

  • Abramowitz, M. and Stegun, I. A. (1965).Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York.

    Google Scholar 

  • Bhattacharyya, A. (1946). On some analogues of the information and their use in statistical estimation,Sankhyā,8, 1–14.

    MathSciNet  Google Scholar 

  • Bhattacharyya, B. C. (1942). The use of McKay’s Bessel function curves for graduating frequency distributions.Sankhyā,6, 175–182.

    MathSciNet  Google Scholar 

  • Blight, J. N. and Rao, P. V. (1974). The convergence of Bhattacharyya bounds,Biometrika,61, 137–142.

    Article  MathSciNet  Google Scholar 

  • Fend, A. V. (1959). On the attainment of Cramér-Rao and Bhattacharyya bounds for the variance of an estimate,Ann. Math. Statist.,30, 381–388.

    MathSciNet  Google Scholar 

  • Ishii, G. (1976). On the attainment of Bhattacharyya bound,Chinese J. Math.,4, 37–45.

    MathSciNet  Google Scholar 

  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994).Continuous Univariate Distributions, Vol. 1, 2nd ed., Wiley, New York.

    MATH  Google Scholar 

  • Lehmann, E. L. and Casella, G. (1998).Theory of Point Estimation, 2nd ed., Springer, New York.

    MATH  Google Scholar 

  • McKay, A. T. (1932). A Bassel function distribution,Biometrika,24, 39–44.

    Article  Google Scholar 

  • Wijsman, R. A. (1973). On the attainment of the Cramér-Rao lower bound,Ann. Statist.,1, 538–542.

    MathSciNet  Google Scholar 

  • Zacks, S. (1971).The Theory of Statistical Inference, Wiley, New York.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, 305-8571, Ibaraki, Japan

    Hidekazu Tanaka & Masafumi Akahira

Authors
  1. Hidekazu Tanaka
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  2. Masafumi Akahira
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Tanaka, H., Akahira, M. On a family of distributions attaining the Bhattacharyya bound. Ann Inst Stat Math 55, 309–317 (2003). https://doi.org/10.1007/BF02530501

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  • DOI: https://doi.org/10.1007/BF02530501

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