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Minimal sufficiency of order statistics in convex models


Author: Lutz Mattner
Journal: Proc. Amer. Math. Soc. 129 (2001), 3401-3411
MSC (2000): Primary 62B05, 62G30, 28A35
DOI: https://doi.org/10.1090/S0002-9939-01-06006-3
Published electronically: May 10, 2001
MathSciNet review: 1845019
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Abstract | References | Similar Articles | Additional Information

Abstract:

Let $\mathcal{P}$ be a convex and dominated statistical model on the measurable space $(\mathcal{X},\mathcal{A})$, with $\mathcal{A}$ minimal sufficient, and let $n\in\mathbb{N} $. Then $\mathcal{A}^{\otimes n}_{\operatorname{sym}}$, the $\sigma$-algebra of all permutation invariant sets belonging to the $n$-fold product $\sigma$-algebra $\mathcal{A}^{\otimes n}$, is shown to be minimal sufficient for the corresponding model for $n$ independent observations, $\mathcal{P}^n = \left\{P^{\otimes n}:P\in\mathcal{P}\right\}$.

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of $\mathcal{A}^{\otimes n}_{\operatorname{sym}}$.


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  • 1. J. L. Doob, Measure theory, Graduate Texts in Mathematics, vol. 143, Springer-Verlag, New York, 1994. MR 1253752
  • 2. Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR 982264
  • 3. E. Grzegorek, Symmetric 𝜎-fields of sets and universal null sets, Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) Lecture Notes in Math., vol. 945, Springer, Berlin-New York, 1982, pp. 101–109. MR 675273
  • 4. HALMOS, P.R. (1950). Measure Theory. Van Nostrand. Reprinted 1974 by Springer, New York. MR 11:504d
  • 5. Marie-Françoise Le Bihan, Monique Littaye-Petit, and Jean-Luc Petit, Exhaustivité par paire, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1753–A1756 (French). MR 0267670
  • 6. D. Landers, Sufficient and minimal sufficient 𝜎-fields, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 197–207. MR 0322995, https://doi.org/10.1007/BF00536559
  • 7. Harald Luschgy, Sur l’existence d’une plus petite sous-tribu exhaustive par paire, Ann. Inst. H. Poincaré Sect. B (N.S.) 14 (1978), no. 4, 391–398 (1979) (French, with English summary). MR 523218
  • 8. Avi Mandelbaum and Ludger Rüschendorf, Complete and symmetrically complete families of distributions, Ann. Statist. 15 (1987), no. 3, 1229–1244. MR 902255, https://doi.org/10.1214/aos/1176350502
  • 9. L. Mattner, Complete order statistics in parametric models, Ann. Statist. 24 (1996), no. 3, 1265–1282. MR 1401849, https://doi.org/10.1214/aos/1032526968
  • 10. MATTNER, L. (1999). Sufficiency, exponential families, and algebraically independent numbers. Math. Meth. Statist. 8, 397-406. CMP 2000:07
  • 11. MATTNER, L. (2000). Minimal sufficienct statistics in location-scale parameter models. Bernoulli 6, 1121-1134. CMP 2001:07
  • 12. Johann Pfanzagl, Parametric statistical theory, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 1994. With the assistance of R. Hamböker. MR 1291393
  • 13. Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
  • 14. Eberhard Siebert, Pairwise sufficiency, Z. Wahrsch. Verw. Gebiete 46 (1979), no. 3, 237–246. MR 521703, https://doi.org/10.1007/BF00538112
  • 15. TORGERSEN, E. (1965). Minimal sufficiency of order statistics in the case of translation- and scale parameters. Skand. Aktuarietidsskrift 48, 16-21.
  • 16. Erik Torgersen, Comparison of statistical experiments, Encyclopedia of Mathematics and its Applications, vol. 36, Cambridge University Press, Cambridge, 1991. MR 1104437
  • 17. WEYL, H. (1946). The Classical Groups. Their Invariants and their Representations. 2nd. Ed. Princeton University Press, Princeton. MR 1:42c

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Additional Information

Lutz Mattner
Affiliation: Department of Statistics, University of Leeds, Leeds LS2 9JT, United Kingdom
Email: mattner@amsta.leeds.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-01-06006-3
Keywords: Comparison of $\sigma$-algebras, nonparametric models, permutation invariance, symmetric sets
Received by editor(s): November 13, 1999
Received by editor(s) in revised form: March 30, 2000
Published electronically: May 10, 2001
Communicated by: Wei Y. Loh
Article copyright: © Copyright 2001 American Mathematical Society