7.6 Show that, in the following estimation problems, all risk functions are continuous. (a) Estimate with

7.6 Show that, in the following estimation problems, all risk functions are continuous.

(a) Estimate θ with L(θ,δ(x)) = [θ − δ(x)]2, X ∼ N(θ , 1).

(b) Estimate θ with L(θ,δ(x)) = |θ − δ(x)|

2, X ∼ Nr(θ,I ).

(c) Estimate λ with L(λ, δ(x)) =
r i=1 λ−m i (λi − δi(x))2, Xi ∼ Poisson(λi), independent.

(d) Estimate β with L(β, δ(x)) =
r i=1 β−m i (βi − δi(x))2, Xi ∼ Gamma(αi, βi), independent, αi known.