Abstract
The problem of testing multinormality against some alternatives invariant with respect to a subgroup of affine transformations is studied. Using the Laplace expansion for integrals, some approximations to the most powerful invariant (MPI) tests are derived. The cases of bivariate exponential and bivariate uniform alternatives are studied in detail, whereas higher-dimensional extensions are only outlined. Those alternatives are irregular and need a special treatment because of the dependence of their supports on the unknown parameters. It is shown that likelihood ratio (LR) tests are asymptotically equivalent to the MPI tests. A Monte Carlo simulation study shows that the powers of the LR tests are very close to the powers of the MPI tests, even in small samples.
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