K-sign depth: From asymptotics to efficient implementation

Abstract The K -sign depth ( K -depth) of a model parameter θ in a data set is the relative number of K -tuples among its residual vector that have alternating signs. The K -depth test based on K -depth, recently proposed by Leckey et al. (2020), is equivalent to the classical residual-based sign test for K = 2 , but is much more powerful for K ≥ 3 . This test has two major drawbacks. First, the computation of the K -depth is fairly time consuming having a polynomial time complexity of degree K , and second, the test requires knowledge about the quantiles of the test statistic which previously had to be obtained by simulation for each sample size individually. We tackle both of these drawbacks by presenting a limit theorem for the distribution of the test statistic and deriving an (asymptotically equivalent) form of the K -depth which can be computed efficiently. For K = 3 , such a limit theorem was already derived in Kustosz et al. (2016a) by mimicking the proof for U -statistics. We provide here a much shorter proof based on Donsker’s theorem and extend it to any K ≥ 3 . As part of the proof, we derive an asymptotically equivalent form of the K-depth which can be computed in linear time. This alternative and the original implementation of the K-depth are compared with respect to their runtimes and absolute difference.