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