Confidence bounds with minimum risk. Let L(, ) be nonnegative and nonincreasing in its second argument for

Confidence bounds with minimum risk. Let L(θ, θ) be nonnegative and nonincreasing in its second argument for θ < θ, and equal to 0 for θ ≥ θ.

If θ and θ∗ are two lower confidence bounds for θ such that P0{θ ≤ θ

} ≤ Pθ{θ∗ ≤ θ

} for all θ
≤ θ, then EθL(θ, θ) ≤ EθL(θ, θ∗).

[Define two cumulative distribution functions F and F∗ by F(u) = Pθ{θ ≤

u}/Pθ{θ∗ ≤ θ}, F∗(u) = Pθ{θ∗ ≤ u}/Pθ{θ∗ ≤ θ} for u<θ, F(u) = F∗(u)=1 for u ≥ θ. Then F(u) ≤ F∗(u) for all u, and it follows from