Local maximin properties of tests in Gaussian shift experiments

Abstract The local behavior of the power of weighted χ 2 -tests and Bayes tests is studied for simple null hypothesis in Gaussian shift experiments. A second order expansion of the power function is given. This expansion provides a shrinking family of ellipsoids (δ E ) 0<δ<1 so that the power of the weighted χ 2 -test is locally constant on the boundary ∂(δ E ). Approximating the weighted χ 2 -test by a sequence of Bayes tests with priors on ∂(δ E ), the weighted χ 2 -test is shown to be locally maximin in the sense of Giri and Kiefer for the family of restricted alternatives given by the complements of the (δ E ) 0<δ<1 .