Skip to main content
SpringerLink
Log in

Simultaneous equivariant estimation of the parameters of linear models

  • Articles
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Basu, D. (1955): On statistics independent of a complete sufficient statistic, Sankhya, A, 15, pp. 377–380.

    MATH  Google Scholar 

  2. Ferguson, T.S. (1967):Mathematical Statistics—A Decision theoretic Approach, Academic Press, New York, San Francisco, London.

    MATH  Google Scholar 

  3. Javier, R. (1986): A necessary and sufficient condition for an estimator to be optimal, Commun. Statist.—Theo. Meth., 15(5), pp. 1647–1651.

    Article  MATH  Google Scholar 

  4. Kiefer, J. (1957): Invariance, Minimax sequential estimation and continuous time processes, Ann. Math Statist., 28, pp. 573–601.

    Article  MathSciNet  Google Scholar 

  5. Lehmann, E.L. (1983):Theory of Point Estimation, John Wiley & Sons, New York.

    MATH  Google Scholar 

  6. Lehmann, E.L. (1986):Testing Statistical Hypotheses, John Wiley & Sons, New York.

    MATH  Google Scholar 

  7. Peisakoff, M. (1950): Transformation Parameters, Unpublished Thesis, Princeton University, Princeton, N.J.

    Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. 87, Ramamurthy Colony Thiru-Vi-ka Nagar, 600 082, Madras, Tamil Nadu, India

    S. Kalpana Bai

  2. Department of Statistics, Loyola College, 600 034, Madras, Tamil Nadu, India

    T. M. Durairajan

Authors
  1. S. Kalpana Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. M. Durairajan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bai, S.K., Durairajan, T.M. Simultaneous equivariant estimation of the parameters of linear models. Statistical Papers 39, 125–134 (1998). https://doi.org/10.1007/BF02925401

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02925401

Key words and Phrases

AMS Subject Classification

Search

Navigation