Tests for multiple regression based on simplicial depth

A general approach for developing distribution free tests for general linear models based on simplicial depth is applied to multiple regression. The tests are based on the asymptotic distribution of the simplicial regression depth, which depends only on the distribution law of the vector product of regressor variables. Based on this formula, the spectral decomposition and thus the asymptotic distribution is derived for multiple regression through the origin and multiple regression with Cauchy distributed explanatory variables. The errors may be heteroscedastic and the concrete form of the error distribution does not need to be known. Moreover, the asymptotic distribution for multiple regression with intercept does not depend on the location and scale of the explanatory variables. A simulation study suggests that the tests can be applied also to normal distributed explanatory variables. An application on multiple regression for shape analysis of fishes demonstrates the applicability of the new tests and in particular their outlier robustness.