Acta Applicandae Mathematica

, Volume 43, Issue 1, pp 17–42 | Cite as

Spectral methods in linear minimax estimation

  • Hilmar Drygas
Article

Abstract

We consider the minimax-linear estimator in a linear regression model with circular constraints. Two necessary and sufficient conditions for the optimality of an estimator, the socalled left spectral equation and the right spectral equation (Girko spectral equation), are derived. For the special case of a simple maximal eigenvalue and a single eigenspace explicit estimation formulas are derived. These formulas also show some of the shortcomings of the minimax-linear estimator (MILE). Finally, the relation with Bayesian analysis and the Hoffmann-Läuter estimator is outlined.

Mathematics Subject Classification (1991)

62J05 

Key words

linear minimax estimation Bayesian approach spectral methods spectral equations Hoffmann-Laüter representation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baksalary J. K. and Kala R.: 1981, Linear transformations preserving best linear unbiased estimators in a general Gauss-Markov model,Ann. Statist. 9, 913–916.Google Scholar
  2. Drygas H.: 1983, Sufficiency and completeness in the general Gauss-Markov model,Sankkyā, A 45(1), 88–98.Google Scholar
  3. Drygas, H.: 1991/92, On an extension of the Girko equality in linear minimax estimation. Kasseler Mathematische Schriften 16/91,Proc. Probastat Conference, Bratislava, 26-30.08.1991, pp. 3–10Google Scholar
  4. Drygas H.: 1992, Linear minimax estimation in a convex linear model—a paradoxon in decision theory?Statistics and Decision 10, 63–66.Google Scholar
  5. Drygas H. and Pilz J.: 1996, On the equivalence of spectral theory and Bayesian analysis in minimax linear estimation,Acta. Appl. Math. 43, 43–57 (this issue).Google Scholar
  6. Drygas H. and Läuter H.: 1993, On the representation of the linear minimax estimator in the convex linear model, in T. Caliński and R. Kala (eds),Proc. Internat. Conf. on Linear Statistical Inference, LINSTAT'93, Kluwer Acad. Publ., Dordrecht, 1994, pp. 13–26.Google Scholar
  7. Gaffke N. and Heiligers B.: 1989, Bayes, admissible and minimax linear estimator in linear models with restricted parameter space,Statistics 20, 487–508.Google Scholar
  8. Gaffke, N. and Mathar, R.: 1990a, Linear minimax estimation and related BayesL-optimal design, in B. Fudessteiner, B. Leugenauer and H. J. Skala (eds),Methods of Operations Research 60, Proc. XIII. Symposium on Operations Research, pp. 617–628.Google Scholar
  9. Gaffke N. and Mathar R.: 1990b, On a class of algorithms from experimental design theory, Report Nr. 196, ‘Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung der Deutschen Forschungsgemeinschaft (DFG)’, University of Augsburg, Germany.Google Scholar
  10. Girko V. L.: 1988,Multidimensional Statistical Analysis (in Russian), Nauka, Moscow (English transl., 1995,Statistical Analysis of Observations of Increasing Dimension, Kluwer Acad. Publ., Dordrecht).Google Scholar
  11. Girko V. L.: 1990a,Theory of Random Determinants, Kluwer Acad. Publ., Dordrecht, Boston, London.Google Scholar
  12. Girko V. L.: 1990b, S-estimators,Vychisl. Prikl. Mat. 71, 90–97 (in Russian).Google Scholar
  13. Girko V. L.: 1990,Theory of Empirical Systems of Equations, Lybid, Kiev (in Russian).Google Scholar
  14. Hofmann K.: 1979, Characterization of minimax linear estimators in linear regression,Math. Operations. Statistik 10, 19–26.Google Scholar
  15. Kiefer J.: 1974, General equivalence theory for optimum design (approximate theory),Ann. Statist. 2, 849–879.Google Scholar
  16. Läuter H.: 1975, A minimax linear estimator for linear parameters under restrictions in form of inequalities,Math. Operations. Statistik 6, 689–695.Google Scholar
  17. Majumdar, D.: 1979, Statistical analysis of nonestimable functionals, Thesis submitted to the Indian Statistical Institute in partial fulfilment of the requirement for the award of Doctor of Philosophy, New Delhi.Google Scholar
  18. Pilz J.: 1986, Minimax linear regression estimation with symmetric parameter restrictions,J. Statist. Plann. Inference 13, 297–318.Google Scholar
  19. Pilz J.: 1991,Bayesian Estimation and Experimental Design in Linear Regression Models, (2nd edn), Wiley, Chichester, New York.Google Scholar
  20. Sion M.: 1958, On general minimax theoremus,Pacific J. Math. 8, 171–176.Google Scholar
  21. Stahlecker P.: 1987,A priori Information und Minimax-Schätzung im linearen Regressionsmodel, Math. System in Economics, 108, Athenäum, Frankfurt.Google Scholar
  22. Stahlecker P. and Lauterbach J.: 1987, Approximate linear minimax estimation in linear regression: Theoretical results,Comm. Statist. A 16, 1101–1116.Google Scholar
  23. Stahlecker P. and Lauterbach J.: 1989, Approximate linear minimax estimation in regression. Analysis with ellipsoidal constraints,Comm. Statist. A 18, 2755–2784.Google Scholar
  24. Trenkler G.: 1981,Biased Estimators in the Linear Regression Model. Mathematical Systems in Economics, Verlag Anton Hain Meisenheim GmbH, Königsstein/Taunus and Oelgeschlager, Gunn and Hain, Cambridge, Mass.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Hilmar Drygas
    • 1
  1. 1.Department of MathematicsUniversity of KasselKasselGermany

Personalised recommendations