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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
Published electronically: May 10, 2001
MathSciNet review: 1845019
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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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Additional Information

Lutz Mattner
Affiliation: Department of Statistics, University of Leeds, Leeds LS2 9JT, United Kingdom
MR Author ID: 315405

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