Comparing the mean vectors across different groups is a cornerstone in the realm of multivariate statistics, with quadratic forms commonly serving as test statistics. However, when the overall hypothesis is rejected, identifying specific vector components or determining the groups among which differences exist requires additional investigations. Conversely, employing multiple contrast tests (MCT) allows conclusions about which components or groups contribute to these differences. However, they come with a trade-off, as MCT lose some benefits inherent to quadratic forms. In this paper, we combine both approaches to get a quadratic form based multiple contrast test that leverages the advantages of both. To understand its theoretical properties, we investigate its asymptotic distribution in a semiparametric model. We thereby focus on two common quadratic forms - the Wald-type statistic and the Anova-type statistic - although our findings are applicable to any quadratic form. Furthermore, we employ Monte-Carlo and resampling techniques to enhance the test's performance in small sample scenarios. Through an extensive simulation study, we assess the performance of our proposed tests against existing alternatives, highlighting their advantages.

翻译：比较不同组间的均值向量是多元统计领域的基石，其中二次型常被用作检验统计量。然而，当整体假设被拒绝时，识别具体的向量分量或确定存在差异的组别需要进一步研究。相反，采用多对比检验（MCT）可以直接推断哪些分量或组别导致了这些差异。但这种方法存在权衡，因为MCT会丧失二次型固有的某些优势。本文结合两种方法，提出了一种基于二次型的多对比检验，以兼取二者之长。为理解其理论性质，我们在半参数模型中研究了其渐近分布。我们重点关注两种常见的二次型——Wald型统计量与方差分析型统计量——尽管所得结论适用于任意二次型。此外，我们采用蒙特卡洛与重抽样技术以提升小样本场景下的检验性能。通过大量模拟研究，我们评估了所提检验方法与现有替代方案的性能，并阐明了其优势。