1. Characterizing angular symmetry and regression symmetry
- Author
-
Rousseeuw, Peter J. and Struyf, Anja
- Subjects
- *
PROBABILITY theory , *SYMMETRY (Physics) , *MULTIPLE regression analysis , *MATHEMATICAL combinations - Abstract
Let
P be a general probability distribution onRp , which need not have a density or moments. We investigate the relation between angular symmetry ofP (a.k.a. directional symmetry) and the halfspace (Tukey) depth. WhenP is angularly symmetric about someθ0 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 thatX is a(p−1) -variate r.v. andY is a univariate r.v. such that the joint distribution of(X,Y) is again a totally general probability distribution onRp . The concept of regression symmetry (RS) about a potential fitθ0 means that in eachx the conditional probability of a positive error equals that of a negative error. If a distribution is regression symmetric about someθ0 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 ofY onX , which we use to construct a statistical test for linearity. [Copyright &y& Elsevier]- Published
- 2004
- Full Text
- View/download PDF