Bivariate Sign Tests

Abstract

Abstract 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 leve...