12. 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 4 without any restriction on the decision space. For the present case, let a decision procedure be given by 8(x) = (8(1)(x),oo .,8(n1)(x» where 8(i)(x) is the probability with which decision

d, is taken when x is observed. If T is sufficient and 1J(i)(t) = E[8(i)(X)lt], the procedures 8 and 1J 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).