Note on the Paper "Transformation Groups and Sufficient Statistics" by J. Pfanzagl

Abstract

Pfanzagl (1972) has shown that under suitable regularity conditions a family of probability measures which is generated by a transformation group and which for some sample size greater than one admits a sufficient statistic which is continuous, real-valued, and equivariant, is equivalent to the location parameter family of normal distributions or to a scale parameter family of Gamma distributions. This was proved under the assumption that the transformation group is Abelian. In this not commutativity of the group is replaced by local compactness.