Tests based on equivariant estimators

Let ?⁽¹⁾ and ?⁽²⁾ be location-equivariant estimators of an unknown location parameter μ. It is shown that the test for H₀: μ ≤ μ₀ versus H_A : μ > μ₀ that rejects H₀ if ?⁽¹⁾ is large is uniformly more powerful than the one that rejects H₀ if ?⁽²⁾ is large if and only if ?⁽²⁾ is “more dispersed” than ?⁽¹⁾. A similar result is obtained for tests on scale using the star-shaped ordering. Examples are given.