Question: 21. Confidence bounds withminimumrisk. Let L(J,~) be nonnegative and nonincreasing in its second argument for < (J , and equal to 0 for (J .

21. Confidence bounds withminimumrisk. Let L«(J,~) be nonnegative and nonincreasing in its second argument for < (J , and equal to 0 for (J . If and ~* are two lower confidence bounds for (J such that Po{~ s (J'} s Po{~* s (J '} for all (J ' s (J , then EoL«(J,~) s EoL«(J,!!*). [Define two cumulative distribution functions F and F* by F(u) = Po (~ =:; u}/Po{!!* s (J}, F*(u) = Po{~* s u}/Po{~* s (J} for u < (J , and F(u) = F*(u) = 1 for u (J . Then F(u) s F*(u) for all u, and it follows from Problem 15 that Eo [ L( (J , ~)] = Po { * s (J }f L((J, u) dF( u) s Po {~* s (J} f L( (J, u) dF*( u) = Eo[L( (J ,~*)] .]

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