2.9 For the situation of Example 2.9, show that:

(a) without loss of generality, the restriction θ ∈ [a, b] can be reduced to θ ∈ [−m, m], m > 0.

(b) If  is the prior distribution that puts mass 1/2 on each of the points ±m, then the Bayes estimator against squared error loss is

δ(x¯) = memnx¯ − e−mnx¯

emnx¯ + e−mnx¯ = m tanh(mnx¯).

(c) For m < 1/

√n, max θ∈[−m,m]

R(θ,δ(X¯ )) = max 3 R(−m, δ(X¯ )), R(m, δ(X¯ ))4 and hence, by Corollary 1.6, δ is minimax.