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Parametric and permutation testing for multivariate monotonic alternatives

机译：多元单调替代项的参数和置换测试

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摘要

We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Bio-metrica 46:328-335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465-472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathe-matica 38:151-161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.
机译：我们首先有兴趣在多元正态分布中分别针对两个受限受限选择项测试k个均值向量的同质性。这个问题是Bartholomew的多元扩展（在Bio-metrica 46：328-335，1959b中）和Sasabuchi等人的扩展。 （1983年在Biometrica 70：465-472中）和Kulatunga和Sasabuchi（在九州大学机械科学与工程系，九州大学数学系：Mathe-matica 38：151-161，1984年）中使用双面有序假设。我们研究了两种情况下的测试问题。一种情况是协方差矩阵是已知的，另一种情况是协方差矩阵未知但很普遍。对于已知协方差矩阵的一般情况，使用似然比方法获得检验统计量。当已知的协方差矩阵是公共和对角线时，将得出检验统计量的零分布，并在不同的显着性水平下计算其临界值。还提出了蒙特卡洛研究，以评估测试的功效。针对公共协方差矩阵未知的情况，提出了一种检验统计量。当协方差矩阵完全未知时，由于很难用经典方法计算出该测试问题的准确p值，因此我们首先基于封闭凸锥上的正交投影，提出测试统计量的重新表述，然后确定其p值的上限。我们还提供基于置换方法和相关测试的非参数组合的通用非参数解决方案。

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