51. (i) One-sided equivariant confidence limits. Let () be real-valued, and suppose that for each ()o, the problem of testing () 5 ()o against () > ()o (in the presence of nuisance parameters {;) remains invariant under a group "For further material on these statistics see Kendall (1970); Aiyar, Guillier, and Albers (1979); and books on nonparametric inference.

G90 and that A(80 ) is a UMP invariant acceptance region for this hypothesis at level a . Let the associated confidence sets S(x) = {8: x E A(8)} be one-sided intervals .S(x) = {8: (x) s 8}, and suppose they are equivariant under all G9 and hence under the group G generated by these. Then the lower confidence limits ~(X) are uniformly most accurate equivariant at confidence level 1 - a in the sense of minimizing P9• ,,{~( X) s 8'} for all 8' < 8. (ii) Let XI' . ." Xn be independently distributed as N(t 0 2). The upper confidence limits 0 2 L(X; - X)2jCo of Example 5, Chapter 5, are uniformly most accurate equivariant under the group Xi = X; +

c, - 00 < c < 00. They are also equivariant (and hence uniformly most accurate equivariant) under the larger group Xi = aX; +

c, - 00 <

a, C < 00.