On the consistency of K-sign depth tests

The consistency of the so-called K-sign depth tests is considered. These tests are based on the K-sign depth, which originated from the simplicial regression depth, but is easier to compute. The K-sign depth tests use only the signs of residuals and are equivalent to the classical sign test for K=2. However, K-sign depth tests with K≥3 show a much better power than the classical sign tests in simulation studies. This property is attributed to the consistency of these tests for K=3. After deriving a general condition for consistency, it is shown that this condition is in particular satisfied for several relevant hypotheses in polynomial regression models.