A bivariate test for location based on simplicial depth

Abstract

We translate the simplicial data depth concept of Liu (1990) into a bivariate test of location that is distribution free under the assumption of angular symmetry about zero, without loss of generality. The test statistic is a count of the number of data triangles that contain zero and rejects the null hypothesis for small values of the test statistic. A straightforward method for computing this statistic is provided. The exact null distribution is computed and tabled for n ≤ 15, the asymptotic null distribution is derived and a formula for approximate critical values is provided. Simulations show that the power of the test is comparable to other bivariate sign type tests. The analogous one dimensional distribution free test is found to be equivalent to a two sided one sample sign test. These ideas are then extended to three dimensions, where we show that, surprisingly, the test is not generally distribution free under the null hypothesis.

Keywords:

• Simplicial depth
• data depth
• degenerate U-statistic
• bivariate test
• distribution free

MATH CLASSIFICATION:

• 62G10 nonparametric inference