# How to prove that the permutation of the points are the minimal sufficient statistics for Cauchy distribution?

I see this everywhere that the permutation of the samples $X_{(1)}, ..., X_{(n)}$ is the minimal sufficient statistic for the Cauchy distribution [1]. It is clear that it is a sufficient statistic,but any hint on how to prove the minimality?

## 1 Answer

Hint: Apply the ratio test (e.g. Theorem 6.2.13 in Casella and Berger's "Statistical Inference, Second Edition"), and consider the roots of the polynomial that is the denominator of the joint Cauchy distribution.