S Sufficiency and completeness. Let .X;F; P# W # 2 ‚/ be a statistical model and T W X ! ˙ a statistic with (for simplicity) countable range ˙. T is called sufficient if there exists a family ¹Qs W s 2 ˙º of probability measures on .X;F/ that do not depend on #

and satisfy P#.  jT D s/ D Qs whenever P#.T D s/ > 0. T is called complete if g 0

is the only function g W ˙ ! R such that E#.g ı T / D 0 for all # 2 ‚. Let  be a real characteristic. Show the following.

(a) Rao–Blackwell 1945/47. If T is sufficient, then every unbiased estimator S of  can be improved as follows: Let gS .s/ WD EQs .S/ for s 2 ˙; then the estimator gS ı T is unbiased and satisfies V#.gS ı T /  V#.S/ for all # 2 ‚.

(b) Lehmann–Scheffé 1950. If T is sufficient and complete and gıT is an unbiased estimator of , then g ı T is in fact a best estimator of .