Powerful generalized sign tests based on sign depth

Abstract

The classical sign test usually provides very bad power for certain alternatives. We present a generalization which is similarly easy to comprehend but much more powerful. It is based on K-sign depth, shortly denoted by K-depth. These so-called K-depth tests are motivated by simplicial regression depth, but are not restricted to regression problems. They can be applied as soon as the true model leads to independent residuals with median equal to zero. Moreover, general hypotheses on the unknown parameter vector can be tested. Since they depend only on the signs of the residuals, these test statistics are outlier robust. While the 2-depth test, i.e. the K-depth test for K = 2, is equivalent to the classical sign test, K-depth test with K ≥3 turn out to be more powerful in many applications. As we will briefly discuss, these tests are also related to runs tests. A drawback of the K-depth test is its fairly high computational effort when implemented naively. However, we show how this inherent computational complexity can be reduced. In order to see why K-depth tests with K ≥ 3 are more powerful than the classical sign test, we discuss the asymptotic behaviour of its test statistic for residual vectors with only few sign changes, which is in particular the case for some nonfits the classical sign test cannot reject. In contrast, we also consider residual vectors with alternating signs, representing models that fit the data very well. Finally, we demonstrate the good power of the K-depth tests for quadratic regression.