(i) One-sided equivariant confidence limits. Let be realvalued, and suppose that, for each 0, the problem...

(i) One-sided equivariant confidence limits. Let θ be realvalued, and suppose that, for each θ0, the problem of testing θ ≤ θ0 against

θ>θ0 (in the presence of nuisance parameters ϑ) remains invariant under a group Gθ0 and that A(θ0) is a UMP invariant acceptance region for this hypothesis at level α. Let the associated confidence sets S(x) = {θ : x ∈ A(θ)}

be one-sided intervals S(x) = {θ : θ(x) ≤ θ}, and suppose they are equivariant under all Gθ and hence under the group G generated by these. Then the lower confidence limits θ(X) are uniformly most accurate equivariant at confidence level 1 −α in the sense of minimizing Pθ,ϑ{θ(X) ≤ θ

} for all θ
< θ.
(ii) Let X1,...,Xn be independently distributed as N(ξ, σ2). The upper confidence limits σ2 ≤ (Xi − X¯)
2/C0 of Example 5.5.1 are uniformly most accurate equivariant under the group X

i = Xi +c, −∞ i = aXi +

c, −∞ <

a, c < ∞.