Tests based on equivariant estimators

Let Ť (1) and Ť (2) be location‐equivariant estimators of an unknown location parameter μ. It is shown that the test for H 0 : μ ≤ μ 0 versus H A : μ > μ 0 that rejects H 0 if Ť (1) is large is uniformly more powerful than the one that rejects H 0 if Ť (2) is large if and only if Ť (2) is “more dispersed” than Ť (1) . A similar result is obtained for tests on scale using the star‐shaped ordering. Examples are given.