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

(i) One-sided equivariant confidence limits. Let θ be real-valued, 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, −∞ < c < ∞. They are also equivariant (and hence uniformly most accurate equivariant) under the larger group X

i = a Xi +

c, −∞ <

a, c < ∞.