Permutation and scale invariant one-sided approximate likelihood ratio tests

Suppose X1,X2,…,Xn is a random sample from Np(θ,V). Because the likelihood ratio test (LRT) of H0:θ1=θ2=⋯=θp versus H1-H0 with H1:θi⩾0 for i=1,2,…,p is complicated, several ad hoc tests have been proposed. Tang et al. [(1989). An approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials. Biometrika 76, 577-583] proposed an approximate LRT; Follmann [(1996). A simple multivariate test for one-sided alternatives. J. Amer. Statist. Assoc. 91, 854-861] suggested rejecting H0 if the usual test of H0 versus ∼H0 rejects H0 with significance level 2α and X¯1+⋯+X¯p>0; and Chongcharoen et al. [(2002). Powers of some one-sided multivariate tests with the population covariance matrix known up to a multiplicative constant. J. Statist. Plann. Inference 107, 103-121] modified Follmann's test to include information about the correlation structure in the sum of the sample means. It is desirable for such tests to be permutation and scale invariant. The LRT is permutation and scale invariant and it is not difficult to modify the Follmann-type tests to have these properties. We study modifications of the tests by Tang, Gnecco and Geller that are permutation and scale invariant.