Instead of conditioning the confidence sets S(X) on a set C, consider a randomized procedure

Instead of conditioning the confidence sets θ ∈ S(X) on a set C, consider a randomized procedure which assigns to each point x a probability

ψ(x) and makes the confidence statement θ ∈ S(x) with probability ψ(x) when x is observed.7

(i) The randomized procedure can be represented by a nonrandomized conditioning set for the observations (X, U), where U is uniformly distributed on (0, 1) and independent of X, by letting C = {(x, u) : u<ψ(x)}.

(ii) Extend the definition of relevant and semirelevant subsets to randomized conditioning (without the use of U).

(iii) Let θ ∈ S(X) be equivalent to the statement X ∈ A(θ). Show that ψ is positively biased semirelevant if and only if the random variables ψ(X) and IA(θ)(X) are positively correlated, where IA denotes the indicator of the set A.