Order statistics from exchangeable random variables are always sufficient

TL;DR

This paper proves that for exchangeable random variables, the distribution of the entire vector conditioned on its order statistics is independent of the original distribution function, highlighting a fundamental property of exchangeability.

Contribution

It establishes that the conditional distribution of exchangeable variables given their order statistics is distribution-independent, revealing a key invariance property.

Findings

Conditional distribution given order statistics is distribution-free.

Order statistics fully determine the distribution of exchangeable variables.

The result applies universally to all exchangeable vectors.

Abstract

Let $(X_1,\ldots,X_n)$ be an exchangeable random vector with distribution function $F$, and denote by $Y_1\leq \cdots\leq Y_n$ the corresponding order statistics. We show that the conditional distribution of $(X_1,\ldots,X_n)$ given $(Y_1,\ldots,Y_n)$ does not depend on $F$.