A simple multivariate test for one-sided alternatives

A simple test for multivariate normal data is proposed that has good power for alternatives where the mean vector tends to be positive; The test rejects if the quadratic form of the sample mean vector exceeds its 2 alpha critical value and the sum of the elements of the mean vector exceeds zero. The proposed test is shown to have type I error rare equal to alpha whether or not the covariance matrix is known. Tight bounds on the power of the proposed test are provided, and the test is compared to three likelihood ratio tests.