The multivariate nonparametric Behrens-Fisher problem
The multivariate nonparametric Behrens-Fisher problem
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DOI：
10.1016/s0378-3758(02)00269-0
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发表时间：
2002-11-01
影响因子：
0.9
通讯作者：
Puri, ML
中科院分区：
文献类型：
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作者：
Brunner, E;Munzel, U;Puri, ML
在本文中，我们考虑了所谓的非参数Beynolds-Fisher问题的多变量情况下，两个样本的独立的多变量观测和平等的边缘分布函数的假设下，在这两个群体是没有假设。此外，我们不要求边缘分布函数的连续性，因此该模型涵盖了具有关系的数据，特别是多变量有序分类数据。一个多变量的相对治疗效果的定义，可以通过使用每个组件内的观察值的中间行列估计，我们推导出这个估计量的渐近分布。此外，估计了估计的相对处理效应的中心向量的未知渐近协方差矩阵，并证明了其L-相合性。为了检验无治疗效应的假设，我们考虑Wald型统计量的秩版本（如Puri和Sen，多变量分析中的非参数方法，Wiley，纽约，1971年）和Brunner等人提出的ANOVA型统计量的秩版本。[J.美国，国家统计局。Assoc.92（1997）1494-1502]，用于单变量非参数模型。模拟结果表明，方差分析型统计量似乎保持相当准确的测试的预先分配的水平（即使是相当小的样本量），而沃尔德型统计量导致或多或少的自由决定。关于功率，两个统计数据都没有一致地优于另一个上级。(C)2002 Elsevier Science B. V.保留所有权利。
In this paper, we consider the multivariate case of the so-called nonparametric Behrens-Fisher problem where two samples with independent multivariate observations are given and the equality of the marginal distribution functions under the hypothesis in the two groups is not assumed. Moreover, we do not require the continuity of the marginal distribution functions so that data with ties and, particularly, multivariate-ordered categorical data are covered by this model. A multivariate relative treatment effect is defined which can be estimated by using the mid-ranks of the observations within each component and we derive the asymptotic distribution of this estimator. Moreover, the unknown asymptotic covariance matrix of the centered vector of the estimated relative treatment effects is estimated and its L-consistency is proved. To test the hypothesis of no treatment effect, we consider the rank version of the Wald-type statistic (as used in Puri and Sen, Nonparametric Methods in Multivariate Analysis, Wiley, New York, 1971) and the rank version of the ANOVA-type statistic which was suggested by Brunner et a]. [J. Amer. Statist. Assoc. 92 (1997) 1494-1502] for univariate nonparametric models. Simulations show that the ANOVA-type statistic appears to maintain the pre-assigned level of the test quite accurately (even for rather small sample sizes) while the Wald-type statistic leads to more or less liberal decisions. Regarding the power, none of the two statistics is uniformly superior to the other. (C) 2002 Elsevier Science B.V. All rights reserved.