Depth-Based Signed-Rank Tests for Bivariate Central Symmetry

Abstract

In this paper, distribution-free, affine invariant, signed-rank test statistics are proposed for the hypothesis that a bivariate distribution is centrally symmetric about an arbitrary specified point. The proposed tests are based on the concept of data depth. However, our tests are inherently orthogonal invariant, an affine invariant version of them is provided by using Tyler’s estimator of scatter. The limiting null distribution of proposed tests is derived and the performance of the proposed tests is evaluated through a Monte Carlo study. This study demonstrates that the tests always detect asymmetry and they are convenient to determine small departures from the null hypothesis with high power. Also it shows that the tests perform well comparing other procedures in the literature.