Null Distribution and Bahadur Efficiency of the Hodges Bivariate Sign Test

Abstract

This note presents a simplified expression for the exact null distribution of the Hodges bivariate sign test. The form is suitable for computing both for small and large sample size and gives the limiting distribution easily. A small table is given for the limiting distribution. The Bahadur limiting efficiency of the test is computed relative to the bivariate Hotelling $T^2$ test for normal alternatives. The value $2/\pi$ is obtained, which is the same as for the one dimensional sign test relative to the $t$-test.