Up to now, powerful outlier robust tests for linear models are based on M-estimators and are quite complicated. On the other hand, the simple robust classical sign test usually provides very bad power for certain alternatives. We present a generalization of the sign test 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. 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 \(K\ge 3\) turn out to be much more powerful in many applications. 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 \(K\ge 3\) are more powerful than the classical sign test, we discuss the asymptotic behavior of its test statistic for residual vectors with only few sign changes, which is in particular the case for some alternatives 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 some examples including high-dimensional multiple regression.

到目前为止，线性模型的强大异常值鲁棒性检验是基于 M 估计量的，并且相当复杂。另一方面，简单的稳健经典符号测试通常为某些替代方案提供非常糟糕的能力。我们提出了符号测试的概括，它同样容易理解但功能更强大。它基于K符号深度，简称为K深度。这些所谓的K深度测试是由单纯回归深度驱动的，但不限于回归问题。一旦真实模型导致中位数为零的独立残差，就可以应用它们。此外，可以测试关于未知参数向量的一般假设。而2-depth测试，即K\(K = 2\)的深度测试，相当于经典的符号测试，使用\( K \ge 3\)的 K 深度测试在许多应用中变得更加强大。K -depth 测试的一个缺点是，在简单实现时计算量相当大。然而，我们展示了如何降低这种固有的计算复杂性。为了了解为什么使用\(K\ge 3\)进行K深度测试比经典符号检验更强大，我们讨论了其检验统计量对于只有很少符号变化的残差向量的渐近行为，尤其是对于经典符号检验不能拒绝的某些替代方案。相比之下，我们还考虑具有交替符号的残差向量，表示非常适合数据的模型。最后，我们展示了K深度测试对一些示例（包括高维多元回归）的强大功效。