Anderson studied the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric convex set. Restricting attention to elliptically contoured, unimodal densities, Mukerjee, Robertson and Wright weakened the assumption of symmetry on the set and obtained monotonicity properties of power functions, including unbiasedness, for some likelihood ratio tests in order restricted inference for the variance-known case. For elliptically contoured, unimodal densities, we weaken the assumption of convexity to obtain similar results in the case of unknown variances. The results apply to situations in which the null hypothesis is a linear space and the alternative is a closed, convex cone.
"Monotonicity Properties of the Power Functions of Likelihood Ratio Tests for Normal Mean Hypotheses Constrained by a Linear Space and a Cone." Ann. Statist. 22 (3) 1547 - 1554, September, 1994. https://doi.org/10.1214/aos/1176325642