4.14 A natural extension of risk domination under a particular loss is to risk domination under a class of losses. Hwang (1985) defines universal domination of δ by δ if the inequality EθL(|θ − δ

(X)|) ≤ EθL(|θ − δ(X)|) for all θ

holds for all loss functions L(·) that are nondecreasing, with at least one loss function producing nonidentical risks.

(a) Show that δ universally dominates δ if and only if it stochastically dominates δ, that is, if and only if Pθ (|θ − δ

(X)| > k) ≤ Pθ (|θ − δ(X)| > k)

for all k and θ with strict inequality for some θ .

[Hint: For a positive random variable Y , recall that EY = ∞

0 P(Y >t) dt. Alternatively, use the fact that stochastic ordering on random variables induces an ordering on expectations. See Lemma 1, Section 3.3 of TSH2.]

(b) For X ∼ Nr(θ , I ), show that the James-Stein estimator δc(x) = (1 − c/|x|

2)x does not universally dominate x. [From (a), it only need be shown that Pθ (|θ −δc(X)| >

k) > Pθ (|θ − X| > k) for some θ and k. Take θ = 0 and find such a k.]