Minimal sufficiency of order statistics in convex models

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 $\mathcal {A}^{\otimes n}_{\operatorname {sym}}$.