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Acta Applicandae Mathematica

, Volume 43, Issue 1, pp 87–95 | Cite as

A subclass of bayes linear estimators that are minimax

  • Kurt Hoffmann
Article

Abstract

In the linear regression model with ellipsoidal parameter constraints, the problem of estimating the unknown parameter vector is studied. A well-described subclass of Bayes linear estimators is proposed in the paper. It is shown that for each member of this subclass, a generalized quadratic risk function exists so that the estimator is minimax. Moreover, some of the proposed Bayes linear estimators are admissible with respect to all possible generalized quadratic risks. Also, a necessary and sufficient condition is given to ensure that the considered Bayes linear estimator improves the least squares estimator over the whole ellipsoid whatever generalized risk function is chosen.

Mathematics Subject Classifications (1991)

primary: 62J05 secondary: 62C10 62C20 

Key words

linear regression Bayes linear estimation restricted parameter space minimax property 

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References

  1. Alson P.: 1988, Minimax properties for linear estimators of the location parameter of a linear model,Statistics 19, 163–171.Google Scholar
  2. Bunke O.: 1975a, Minimax linear, ridge and shrunken estimators for linear parameters,Math. Operations. Statist., Ser. Statist. 6, 697–701.Google Scholar
  3. Bunke O.: 1975b, Improved inference in linear models, with additional information,Math. Operations. Statist., Ser. Statist. 6, 817–829.Google Scholar
  4. Gaffke N. and Heiligers B.: 1989, Bayes, admissible and minimax linear estimators in linear models with restricted parameter space,Statistics 20, 487–508.Google Scholar
  5. Gaffke N. and Heiligers B.: 1991, Note on a paper by P. Alson,Statistics 22, 3–8.Google Scholar
  6. Hoffmann K.: 1979, Characterization of iminimax linear estimators in linear regression,Math. Operations. Statist., Ser. Statist. 10, 19–26.Google Scholar
  7. Hoffmann K.: 1980, Admissible improvements of the least squares estimator,Math. Operations. Statist., Ser. Statist. 11, 373–388.Google Scholar
  8. Kuks J. and Olman V.: 1971, A minimax estimator of regression coefficients (in Russian),Izv. Akad. Nauk Eston. SSR 20, 480–482.Google Scholar
  9. Läuter H.: 1975, A minimax linear estimator for linear parameters under restrictions in form of inequalities,Math. Operations. Statist., Ser. Statist. 6, 689–695.Google Scholar
  10. Pilz J.: 1986, Minimax linear regression estimation with symmetric parameter restrictions,J. Statist. Plann. Inference 13, 297–318.Google Scholar
  11. Rao C. R.: 1976, Estimation of parameters in a linear model,Ann. Statist. 4, 1023–1037.Google Scholar
  12. Schipp B., Trenkler G. and Stahlecker P.: 1988, Minimax estimation with additional linear restrictions—a simulation study,Comm. Statist. Simula. 17, 393–406.Google Scholar
  13. Stahlecker P. and Trenkler G.: 1989, Full and partial minimax estimation in regression analysis with additional linear constraints,Linear Algebra Appl.111, 279–292.Google Scholar
  14. Trenkler G. and Stahlecker P.: 1987, Quasi minimax estimation in the linear regression model,Statistics 18, 219–226.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Kurt Hoffmann
    • 1
  1. 1.Ludwig-Renn-Strasse 37BerlinGermany

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