Thanks to its favorable properties, the multivariate normal distribution is still largely employed for modeling phenomena in various scientific fields. However, when the number of components $p$ is of the same asymptotic order as the sample size $n$, standard inferential techniques are generally inadequate to conduct hypothesis testing on the mean vector and/or the covariance matrix. Within several prominent frameworks, we propose then to draw reliable conclusions via a directional test. We show that under the null hypothesis the directional $p$-value is exactly uniformly distributed even when $p$ is of the same order of $n$, provided that conditions for the existence of the maximum likelihood estimate for the normal model are satisfied. Extensive simulation results confirm the theoretical findings across different values of $p/n$, and show that the proposed approach outperforms not only the usual finite-$p$ approaches but also alternative methods tailored for high-dimensional settings.
翻译:由于其有利的特性,多变量正常分布在很大程度上仍然用于在各科学领域进行模型化现象,然而,如果成分数量与样本大小相同,但美元与美元相同,标准推断技术通常不足以对平均矢量和/或共变矩阵进行假设测试。在若干突出的框架内,我们随后提议通过方向性测试得出可靠结论。我们表明,在无效假设下,方向值$p$-价值的分布完全一致,即使美元为10美元,只要满足正常模型存在最大可能性估计数的条件。广泛的模拟结果证实了不同数值的理论结论,即$/n美元,并表明拟议方法不仅优于通常的定额-p美元方法,而且优于适合高维环境的替代方法。