7. Let x = (XI""'Xn ) , and let ge(x,~) be a family of probability densities depending on 8 = (81" • • , 8r ) and the real parameter ~, and jointly measurable in x and For each 8, let he(~) be a probability density with respect to a a-finite measure II such that Pe(x) = !ge(x, ~)hea) dll(~) exists. We shall say that a function I of two arguments U = (uI , ..• , ur ) , v = (VI" ' " Vs) is nondecreasing in (u , v) if [tu', v)/I(u, v) [t u', v')/I(u, v') for all (u, v) satisfying u, ut, vj V; (i = 1, .. . , r; j = 1, . .. ,s). Then Pe(x) is nondecreasing in (x, 8) provided the product ge(x, ~)he(n is

(a) nondecreasing in (x,8) for each fixed ~;

(b) nondecreasing in (8,~) for each fixed x ;

(c) nondecreasing in (x ,~) for each fixed 8. [Interpreting ge(x, ~) as the conditional density of x given t and he(~) as the a priori density of t let p(~) denote the a posteriori density of given x, and let p'a) be defined analogously with 8' in place of 8. That Pe(x) is nonde-creasing in its two arguments is equivalent to fgo(x "~) p(n d,,(n s fgO,(x',n p'(n d,,(n · go(x,n go ,(x,n By

(a) it is enough to prove that f go(x', n , D = (t) [p (~) - p(nJ d,,(~) O. go x , .. 531 Let L = {~: p'a)/p(~) < I} and S+ = {~ p'a)/p(~) I} . By

(b) the set S _ lies entirely to the left of S+. It follows from

(c) that there exists as b such that D= af. [p'(n - p(nJ d,,(~) + bf. [p'(n - p(~)J d,,(n, s_ s+ and hence that D = (b -

a) f. [p '(~) - pa») d"a) 0.