A Conditionally Distribution-Free Multivariate Sign Test for One-Sided Alternatives

We consider the problem of testing the hypothesis that a multivariate location vector is in the positive orthant. A conditionally distribution-free sign test is proposed for this problem. This test is related to the Hodges test and can be motivated by the union-intersection principle. Moreover, it is valid under very mild assumptions. A characterization of the conditional null distribution of the test statistic is given. We provide a step-by-step procedure that can be used to perform the test in practice. In the bivariate case, an explicit formula for the exact null conditional distribution of the test statistic is derived. This conditional distribution can be used to compute exact conditional P values. A simulation study compares the new test to some competitors, including the likelihood ratio test. The results show that the new test is very competitive for a wide variety of distributional models. A real data example illustrating the use of the test is also presented.