Statistical Papers

, Volume 39, Issue 2, pp 125–134 | Cite as

Simultaneous equivariant estimation of the parameters of linear models

  • S. Kalpana Bai
  • T. M. Durairajan


Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.

Key words and Phrases

Convex loss function equivariant estimators characterization linear model regression parameters 

AMS Subject Classification

Primary 62A05 Secondary 62J05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Basu, D. (1955): On statistics independent of a complete sufficient statistic, Sankhya, A, 15, pp. 377–380.MATHGoogle Scholar
  2. 2.
    Ferguson, T.S. (1967):Mathematical Statistics—A Decision theoretic Approach, Academic Press, New York, San Francisco, London.MATHGoogle Scholar
  3. 3.
    Javier, R. (1986): A necessary and sufficient condition for an estimator to be optimal, Commun. Statist.—Theo. Meth., 15(5), pp. 1647–1651.MATHCrossRefGoogle Scholar
  4. 4.
    Kiefer, J. (1957): Invariance, Minimax sequential estimation and continuous time processes, Ann. Math Statist., 28, pp. 573–601.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Lehmann, E.L. (1983):Theory of Point Estimation, John Wiley & Sons, New York.MATHGoogle Scholar
  6. 6.
    Lehmann, E.L. (1986):Testing Statistical Hypotheses, John Wiley & Sons, New York.MATHGoogle Scholar
  7. 7.
    Peisakoff, M. (1950): Transformation Parameters, Unpublished Thesis, Princeton University, Princeton, N.J.Google Scholar
  8. 8.
    Pitman, E.J.G. (1939): The estimation of location and scale parameters of a continuous population of any given form, Biometrika, 39, pp. 391–421.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • S. Kalpana Bai
    • 1
  • T. M. Durairajan
    • 2
  1. 1.MadrasIndia
  2. 2.Department of StatisticsLoyola CollegeMadrasIndia

Personalised recommendations