Initially developed in Brunner et al. (1997), the Anova-type-statistic (ATS) is one of the most used quadratic forms for testing multivariate hypotheses for a variety of different parameter vectors $\boldsymbol{\theta}\in\mathbb{R}^d$. Such tests can be based on several versions of ATS and in most settings, they are preferable over those based on other quadratic forms, as for example the Wald-type-statistic (WTS). However, the same null hypothesis $\boldsymbol{H}\boldsymbol{\theta}=\boldsymbol{y}$ can be expressed by a multitude of hypothesis matrices $\boldsymbol{H}\in\mathbb{R}^{m\times d}$ and corresponding vectors $\boldsymbol{y}\in\mathbb{R}^m$, which leads to different values of the test statistic, as it can be seen in simple examples. Since this can entail distinct test decisions, it remains to investigate under which conditions tests using different hypothesis matrices coincide. Here, the dimensions of the different hypothesis matrices can be substantially different, which has exceptional potential to save computation effort. In this manuscript, we show that for the Anova-type-statistic and some versions thereof, it is possible for each hypothesis $\boldsymbol{H}\boldsymbol{\theta}=\boldsymbol{y}$ to construct a companion matrix $\boldsymbol{L}$ with a minimal number of rows, which not only tests the same hypothesis but also always yields the same test decisions. This allows a substantial reduction of computation time, which is investigated in several conducted simulations.

翻译：最初由 Brunner 等人（1997 年）提出的方差分析型统计量（ATS）是用于检验多种不同参数向量 $\boldsymbol{\theta}\in\mathbb{R}^d$ 的多元假设时最常用的二次型之一。此类检验可基于多个版本的 ATS 进行，且在多数设定下，它们优于基于其他二次型（例如 Wald 型统计量，WTS）的检验。然而，同一个原假设 $\boldsymbol{H}\boldsymbol{\theta}=\boldsymbol{y}$ 可以通过多种假设矩阵 $\boldsymbol{H}\in\mathbb{R}^{m\times d}$ 及对应的向量 $\boldsymbol{y}\in\mathbb{R}^m$ 来表达，这会导致检验统计量的取值不同，正如简单示例中所见。由于这可能引发不同的检验决策，因此仍需探究在何种条件下使用不同假设矩阵的检验会得到一致结果。此处，不同假设矩阵的维度可能存在显著差异，这为节省计算量提供了重要潜力。在本文中，我们证明对于方差分析型统计量及其某些变体，可以为每个假设 $\boldsymbol{H}\boldsymbol{\theta}=\boldsymbol{y}$ 构造一个具有最少行数的伴随矩阵 $\boldsymbol{L}$，该矩阵不仅能检验相同的假设，而且总能产生相同的检验决策。这可以显著减少计算时间，这一点在多项模拟实验中得到验证。