Characterizing angular symmetry and regression symmetry

Let P be a general probability distribution on , which need not have a density or moments. We investigate the relation between angular symmetry of P (a.k.a. directional symmetry) and the halfspace (Tukey) depth. When P is angularly symmetric about some θ0 we derive the expression of the maximal Tukey depth. Surprisingly, the converse also holds, hence angular symmetry is completely characterized by Tukey depth. This fact puts some existing tests for centrosymmetry and for uniformity of a directional distribution in a new perspective. In the multiple regression framework, we assume that X is a (p−1)-variate r.v. and Y is a univariate r.v. such that the joint distribution of (X,Y) is again a totally general probability distribution on . The concept of regression symmetry (RS) about a potential fit θ0 means that in each x the conditional probability of a positive error equals that of a negative error. If a distribution is regression symmetric about some θ0 then the maximal regression depth has a certain expression. It turns out that the converse holds as well. Therefore, regression depth characterizes the linearity of the conditional median of Y on X, which we use to construct a statistical test for linearity.status: publishe