Advertisement

The lower bound for the variance of unbiased estimators for one-directional family of distributions

  • Masafumi Akahira
  • Kei Takeuchi
Article

Summary

In this paper we introduce the concept of one-directionality which includes both cases of location (and scale) parameter and selection parameter and also other cases, and establish some theorems for shapr lower bounds and for the existence of zero variance unbiased estimator for this class of non-regular distributions.

Key words and phrases

Cramér-Rao bound Bhattacharyya bound unbiased estimator one-directional family of distributions sharp lower bound 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Akahira, M., Puri, M. L. and Takeuchi, K. (1984). Bhattacharyya bound of variances of unbiased estimators in non-regular cases,Ann. Inst. Statist. Math.,38, 35–44.CrossRefGoogle Scholar
  2. [2]
    Chapman, D. G. and Robbins, H. (1951). Minimum variance estimation without regularity assumptions,Ann. Math. Statist.,22, 581–586.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Fraser, D. A. S. and Guttman, I. (1952). Bhattacharyya bounds without regularity assumptions,Ann. Math. Statist.,23, 629–632.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Kiefer, J. (1952). On minimum variance in non-regular estimation,Ann. Math. Statist.,23, 627–629.CrossRefGoogle Scholar
  5. [5]
    Móri, T. F. (1983). Note on the Cramér-Rao inequality in the non-regular cases: The family of uniform distributions,J. Statist. Plann. Inference,7, 353–358.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Morimoto, H. and Sibuya, M. (1967). Sufficient statistics and unbiased estimation of restricted selection parameter,Sankhyã, A27, 15–40.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Takeuchi, K. and Akahira, M. (1983). A note on minimum variance,Metrika,33, 85–91.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Víncze, I. (1979). On the Cramér-Fréchet-Rao inequality in the non-regular case, InContributions to Statistics, the Jaroslav Hájek Memorial Volume, Academia, Prague, 253–262.CrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1987

Authors and Affiliations

  • Masafumi Akahira
    • 1
    • 2
  • Kei Takeuchi
    • 1
    • 2
  1. 1.University of Electro-CommunicationsTokyo
  2. 2.University of TokyoTokyoJapan

Personalised recommendations