Sufficiency and invariance (Q1063959)

Suppose that a statistical decision problem is invariant under a group of transformations \(g\in G\). T(X) is equivariant if there exists \(g^*\in G^*\) such that \(T(g(X))=g^*(T(X))\). We show that the minimal sufficient statistic is equivariant and that if T(X) is an equivariant sufficient statistic and d(X) is invariant under G, then \(d^*(T)=Ed(X)| T\) is invariant under \(G^*\).