7.5 A decision problem is monotone (as defined by Karlin and Rubin 1956; see also Brown, Cohen and Strawderman 1976 and Berger 1985, Section 8.4) if the loss function L(θ,δ) is, for each θ, minimized at δ = θ and is an increasing function of |δ − θ|. An estimator δ is monotone if it is a nondecreasing function of x.

(a) Show that if L(θ,δ) is convex, then the monotone estimators form a complete class.

(b) If δ(x) is not monotone, show that the monotone estimator δ defined implicitly by Pt(δ

(X) ≤ t) = Pt(δ(X) ≤ t) for every t satisfies R(θ,δ

) ≤ R(θ,δ) for all θ.

(c) If X ∼ N(θ , 1) and L(θ,δ)=(θ − δ)

2, construct a monotone estimator that dominates

δa (x) =







−2a − x if x < −a x if |x| ≤ a 2a − x if x > a.