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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:

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
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

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