Generalized sign tests based on sign depth

Abstract

We introduce generalized sign tests based on K-sign depth, shortly denoted by K-depth. These so-called K-depth tests are motivated by simplicial regression depth. Since they depend only on the signs of the residuals, these test statistics are easy to comprehend and outlier robust. We show that the K-depth test with K = 2 is equivalent to the classical sign test so that K-depth tests with K > 2 are generalizations of the classical sign test. Since the K-depth test with K = 2 is equivalent to the classical sign test, it has the same drawbacks as the classical sign test. However, the generalized sign tests with K > 2 are much more powerful. We show this by deriving their behavior at observations with few sign changes. Thereby we also prove an upper bound for the K-depth which is attained by observations with alternating signs of residuals. Furthermore, we prove the consistency of the K- depth. Finally, we demonstrate the good power of the K-depth tests for relevance testing, quadratic regression, and tests for explosive AR(2) and nonlinear AR(1) regression.