Tests based on sign depth for multiple regression

The extension of simplicial depth to robust regression, the so-called simplicial regression depth, provides an outlier robust test for the parameter vector of regression models. Since simplicial regression depth often reduces to counting the subsets with alternating signs of the residuals, this led recently to the notion of sign depth and sign depth test. Thereby sign depth tests generalize the classical sign tests. Since sign depth depends on the order of the residuals, one generally assumes that the D-dimensional regressors (explanatory variables) can be ordered with respect to an inherent order. While the one-dimensional real space possesses such a natural order, one cannot order these regressors that easily for D > 1 because there exists no canonical order of the data in most cases. For this scenario, we present orderings according to the Shortest Hamiltonian Path and an approximation of it. We compare them with more naive approaches like taking the order in the data set or ordering on the basis of a single quantity of the regressor. The comparison bases on the computational runtime, stability of the order when transforming the data, as well as on the power of the resulting sign depth tests for testing the parameter vector of different multiple regression models. Moreover, we compare the power of our new tests with the power of the classical sign test and the F-test. Thereby, the sign depth tests based on our distance based approaches show similar power as the F-test for normally distributed residuals with the additional benefit of being much more robust against outliers