Generalized sign tests based on sign depth
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Date
2018
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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.
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Keywords
simplicial regression depth, AR(2) regression, nonlinear AR(1) regression, quadratic regression, relevance test, sign test, K-depth test, K-sign depth