Bivariate Tests for Location and Their Bahadur Efficiencies

Abstract

We consider $(X_1, Y_1), (X_2, Y_2), \cdots, (X_n, Y_n)$, random sample from an absolutely continuous bivariate, population with symmetric density $f(x, y)$ and test $H_0: f(x, y)$ symmetric about (0,0) against $H_1:$ all possible location alternatives. Hotelling's $T^2$ statistic is often used for this test. We denote a form of this statistic by $T_n^{(2)}$ and make an exact Bahadur efficiency comparison of $T_n^{(2)}$ with respect to three of its competitors: a new bivariate Wilcoxon signed rank test $T_n^{(1)}$, Hodges' bivariate sign test $T_n^{(3)}$, and Blumen's bivariate sign test $T_n^{(4)}$. When a bivariate normal alternative with parameter $\Delta = \mu'\sum^{-1}\mu$ obtains, it is shown that the exact Bahadur slopes of $T_n^{(1)}, T_n^{(2)}$, and $T_n^{(3)}$ are identical to the exact slopes of their univariate analogues with a univariate normal alternative with parameter $\Delta = \mu/\sigma$ obtains. In this case, the exact Bahadur efficiency of $T_n^{(1)}$ is uniformly better than either the exact Bahadur efficiency of $T_n^{(3)}$ or $T_n^{(4)}$ with respect to $T_n^{(2)}$.