Testing axial symmetry by means of directional regression quantiles

Abstract

The article describes how directional quantiles can be useful for testing the null hypothesis that a multivariate distribution is symmetric around a line in a given direction. It also generalizes the proposed tests to residual distributions in a linear regression setup, discusses their use for statistical inference regarding equality of distributions, equality of scale, or exchangeability, and illustrates the achievements with carefully designed pictures and examples.

Testing exchangeability of multivariate distributions

“…It was recently suggested to replace testing the exchangeability of a distribution by testing axial symmetry, e.g. using the test based on (multivariate) directional quantiles of [ 19 ]; it namely holds that if a multivariate distribution is exchangeable after a shift, then it is symmetric around the axis of the first orthant. Another connection mentioned in [ 20 ] is related to p independent univariate distributions, which are assumed to be the same up to their location; their joint distribution (after a shift) has to be exchangeable.…”

“…Thus, the test whether their distributions are the same (up to a shift) may be performed as a test of exchangeability of the joint distribution (again up to a shift). Other useful relationships between exchangeability of a distribution and some symmetry concepts (defined in [ 33 , 39 ]) were also described in [ 19 , 20 ].…”