Bivariate Sign Tests Based on the Sup, L1 and L2 Norms

The bivariate location problem is considered. The sup, L1 and L2 norms are used to construct bivariate sign tests from the univariate sign statistics computed on the projected observations on all lines passing through the origin. The tests so obtained are affine-invariant and distribution-free under the null hypothesis. The sup-norm gives rise to Hodges' test. A class of tests derived from the L2-norm, with Blumen's test as a member, is seen to be related to a class proposed by Oja and Nyblom (1989, J. Amer. Statist. Assoc., 84, 249-259). The L1-norm gives rise to a new test. Its asymptotic null distribution is seen to be the same as that of the L1-norm of a certain normal process related to the standard Wiener process. An explicit expression of its cumulative distribution function is given. A simulation study will examine the merits of the three approaches.