5.26 Kiefer inequality. (a) Let X have density (with respect to ) p(x,) which is > 0...

5.26 Kiefer inequality.

(a) Let X have density (with respect to µ) p(x,θ) which is > 0 for all x, and let 1 and 2 be two distributions on the real line with finite first moments. Then, any unbiased estimator δ of θ satisfies var(δ) ≥ [

Wd1(W) − Wd2(W)]2

ψ2(x,θ)p(x,θ) dµ(x)

where

ψ(x,θ) =

θ p(x,θ + W)[d1(W) − d2(W)]

p(x,θ)

with θ = {W : θ + Wε }.

(b) If 1 and 2 assign probability 1 to W = 0 and W, respectively, the inequality reduces to (5.6) with g(θ) = θ. [Hint: Apply (5.1).] (Kiefer 1952.)