Most Powerful Invariant Tests for Binormality

Abstract

We investigate the problem of testing multinormality against alternatives invariant with respect to the affine group of transformations $G$ and against left-bounded alternatives defined by Szkutnik. The last problem remains invariant under a suitably chosen subgroup $G^\ast$ of $G$. Using Wijsman's theorem we find general forms of the most powerful $G$- and $G^\ast$-invariant tests for multinormality which opens the way to an extension of the one-dimensional results of Uthoff to the bivariate case. We find explicit forms of tests against bivariate exponential and bivariate uniform alternatives. A Monte Carlo approximation of the power of these tests is given. This provides us with upper bounds for the power of all invariant tests for binormality against the alternatives considered. The maximum property of the tests obtained is also studied.