Bivariate Sign Tests

In this article the bivariate location problem is treated. New appealing bivariate analogs of the univariate sign tests are proposed for testing the null hypothesis concerning the unknown symmetry center. These tests remain unaltered under any nonsingular linear transformation. From these promising findings a whole family of locally most powerful invariant sign tests is introduced. The tests proposed earlier (Blumen 1958; Hodges 1955) are specific members of this family. For example, Blumen's test appears to be optimal against bivariate normal (or any other elliptic) alternatives. The limiting distributions are derived both under the null hypothesis and under the contiguous alternatives. These limiting distributions are then used to derive asymptotic relative efficiencies. It is found that Blumen's test has the efficiency .785 relative to Hotelling's test against bivariate normal alternatives. For other locally most powerful sign tests the corresponding efficiency depends on the significance level and the power, but not too strongly. In fact, the value .785 also serves as an approximation for other sign tests. The lower bound for the efficiency of Blumen's test relative to Hotelling's test is established against elliptic alternatives. The restriction to unimodal elliptic alternatives increases the lower bound to ½. Finally, some results on Hodges's test are included. Bivariate sign tests can be applied, for example, in paired comparison situations with a bivariate response variable. The hypothesis of no difference between two treatments then implies that the paired response differences are symmetric about the origin, which can be tested by using bivariate sign tests. No extra assumptions concerning the unknown bivariate distributions are needed. In addition, other examples of applications are included. The proposed tests use the direction angles of the observations. Therefore, they can be used also in testing the uniformity (and a kind of symmetry) of a circular distribution. In Tables 1, 2, and 3 critical values for three different sign tests are given.