Anderson (1955) established the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric, convex set. A similar result is developed for integrals of elliptically contoured, unimodal densities over translates of an asymmetric, convex set in certain directions related to the set. This result is used to establish some monotonicity properties of the power functions of the likelihood ratio tests for determining whether or not a vector of normal means satisfies a specified ordering.
"A Probability Inequality for Elliptically Contoured Densities with Applications in Order Restricted Inference." Ann. Statist. 14 (4) 1544 - 1554, December, 1986. https://doi.org/10.1214/aos/1176350175