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Gamma-minimax estimators in the exponential family

Seria
Rozprawy Matematyczne tom/nr w serii: 308 wydano: 1991
Zawartość
Warianty tytułu
Abstrakty
EN
The Γ-minimax estimator under squared error loss for the unknown parameter of a one-parameter exponential family with an unbiased sufficient statistic having a variance which is quadratic in the parameter is explicitly determined for a class Γ of priors consisting of all distributions whose first two moments are within some given bounds. This generalizes the choice of Γ in Jackson et al. (1970) as well as the unrestricted case. It is shown that the underlying statistical game is always strictly determined and that there exists a Γ-minimax estimator which is a linear function of the unbiased sufficient statistic. If the bounds for both prior moments are effective then there exists a least favourable prior in Γ which is a member of the Pearsonian family.
EN

CONTENTS
1. Introduction and summary....................5
2. A class of exponential families..............6
3. The estimation problem......................12
4. Solution of the statistical games.........17
5. Some special cases............................32
6. Concluding remark.............................33
References............................................35
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 308
Liczba stron
35
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCVIII
Daty
wydano
1991
otrzymano
1989-01-25
poprawiono
1990-05-24
Twórcy
  • Department of Mathematics, Tongji University, Si Ping 1239, Shanghai, China
  • An der Neckarspitze 6, D-6900 Heidelberg, Germany
  • Fachbereich Mathematik, Technische Hochschule, Arbeitsgruppe Stochastik und Operations Research, Schloß Gartenstr. 7, D-6100 Darmstadt, Germany
  • Fachbereich Mathematik, Technische Hochschule, Arbeitsgruppe Stochastik und Operations Research, Schloß Gartenstr. 7, D-6100 Darmstadt, Germany
Bibliografia
  • J. O. Berger (1985), Statistical Decision Theory and Bayesian Analysis, Springer, Berlin, 2nd ed.
  • D. Bierlein (1968), The development of the concept of statistical decision theory, Europ. Meet. Amsterdam, Select. Statist. Papers 1, 27-35.
  • D. Bierlein (1967), Zur Einbeziehung der Erfahrung in spieltheoretische Modelle, Operations Res. Verf. 3, 29-54.
  • L. Chen, J. Eichenauer-Herrmann and J. Lehn (1990), Gamma-minimax estimation of a multivariate normal mean, Metrika 37, 1-6.
  • T. A. DeRouen and T. J. Mitchell (1974), A $G_1$-minimax estimator for a linear combination of binomial probabilities, J. Amer. Statist. Assoc. 69, 231-233.
  • P. Diaconis and D. Ylvisaker (1979), Conjugate priors for exponential families, Ann. Statist. 7, 269-281.
  • J. Eichenauer, J. Lehn and S. Rettig (1988), A gamma-minimax result in credibility theory, Insur. Math. Econ. 7, 49-57.
  • K. Fan (1953), Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39, 42-47.
  • D. A. Jackson, T. M. O'Donovan, W. J. Zimmer and J. J. Deely (1970), $G_2$-minimax estimators in the exponential family, Biometrika 57, 439-443.
  • N. L. Johnson and S. Kotz (1970), Continuous Univariate Distributions -1, Houghton Mifflin, Boston.
  • J. Kindler (1981), Statistische Entscheidungsprobleme mit nichtkompaktem Aktionenraum, Manuscripta Math. 34, 255-263.
  • E. L. Lehmann (1983), Theory of Point Estimation, Wiley, New York.
  • C. N. Morris (1982), Natural exponential families with quadratic variance functions, Ann. Statist. 10, 65-80.
  • C. N. Morris (1983), Natural exponential families with quadratic variance functions: statistical theory, Ann. Statist. 11, 515-529.
  • M. Neumann (1977), Bemerkungen zum von Neumannschen Minimaxtheorem, Arch. Math. (Basel) 29, 96-105.
  • H. Robbins (1964), The empirical Bayes approach to statistical decision problems, Ann. Math. Statist. 35, 1-20.
  • F. J. Samaniego (1975), On T-minimax estimation, Amer. Statistician 29, 168-169.
  • D. L. Solomon (1972), Λ-minimax estimation of a multivariate location parameter, J. Amer. Statist. Assoc. 67, 641-646.
  • M. C. K. Tweedie (1967), A mean-square-error characterization of binomial-type distributions, Ann. Math. Statist. 38, 620-623.
  • A. Wald (1950), Statistical Decision Functions, Wiley, New York.
  • A. J. Weir (1973), Lebesgue Integration and Measure, Cambridge Univ. Press.
Języki publikacji
EN
Uwagi
1980 Mathematics Subject Classification: (1985 Revision): Primary 62C99; Secondary 62F10.
Identyfikator YADDA
bwmeta1.element.zamlynska-f53b6f38-8c1f-4764-82f9-f71400ad28a0
Identyfikatory
ISBN
83-85116-05-2
ISSN
0012-3862
Kolekcja
DML-PL
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