A New Bivariate Sign Test

Abstract

A sign test for the hypothesis that the medians of two or more variables (not necessarily independent) have a particular value is described in detail for the bivariate case. The slopes of the vectors from the bivariate median to the n sample points are ordered and we let a i = ±1 according as the vector corresponding to the j th slope is above or below a horizontal line through the median. The bivariate test is based on the statistic v 2 = 2 ( v 1 2 + v 2 2 )/ n , where v 1 [Sgrave] a i cos (π j / n ), v 2 = [Sgrave] a i sin (π j / n ), and the sum is over the integers j = 1, 2, …, n . The statistic has the chi-square distribution for large samples. The power of this test relative to other possible sign tests and Hotelling's T 2 is discussed.