BIVARIATE SIGN TESTS BASED ON THE SUP, L_{1} AND L_{2} NORMS

DENIS LAROCQUE^{1}** , SERGE TARDIF^{1}*** AND CONSTANCE VAN EEDEN^{2}

1Départment de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada H3C 3J7

2Department of Statistics, The University of British Columbia, Vancouver, Canada V6T 1Z2.

(Received June 23, 1997; revised March 19, 1999)

Abstract. The bivariate location problem is considered. The sup, L_{1} and L_{2} 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 L_{2}-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 L_{1}-norm gives rise to a new test. Its asymptotic null distribution is seen to be the same as that of the L_{1}- 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.

Key words and phrases: Location problem, distribution-free, affine-invariance, normal process, Wiener process, L_{1}-norm, L_{2}-norm, Hodges’ test, Blumen’s test.