For a decision problem with a finite number of decisions, the class of procedures depending on a sufficient statistic T only is essentially complete. [For Euclidean sample spaces this follows from Theorem 2.5.1 without any restriction on the decision space. For the present case, let a decision procedure be given by

δ(x) = (δ(1)

(x), . . . , δ(m)

(x)) where δ(i)

(x) is the probability with which decision di is taken when x is observed. If T is sufficient and η(i)

(t) = E[δ(i)

(X) | t], the procedures δ and η have identical risk functions.] [More general versions of this result are discussed, for example, by Elfving (1952), Bahadur (1955), Burkholder

(1961), LeCam (1964), and Roy and Ramamoorthi (1979).]