Caizhu Huang | Searcharxiv

arXiv2026

Directional testing for one-way MANOVA in divergent dimensions

Testing the equality of mean vectors across $g$ different groups plays an important role in many scientific fields. In regular frameworks, likelihood-based statistics under the normality assumption offer a general solution to this task. However, the accuracy of standard asymptotic results is not reliable when the dimension $p$ of the data is large relative to the sample size $n_i$ of each group. We propose here an exact directional test for the equality of $g$ normal mean vectors with identical unknown covariance matrix in a high dimensional setting, provided that $\sum_{i=1}^g n_i \ge p+g+1$. In the case of two groups ($g=2$), the directional test coincides with the Hotelling's $T^2$ test. In the more general situation where the $g$ independent groups may have different unknown covariance matrices, although exactness does not hold, simulation studies show that the directional test is more accurate than most commonly used likelihood{-}based solutions, at least in a moderate dimensional setting in which $p=O(n_i^τ)$, $τ\in (0,1)$. Robustness of the directional approach and its competitors under deviation from the assumption of multivariate normality is also numerically investigated. Our proposal is here applied to data on blood characteristics of male athletes and to microarray data storing gene expressions in patients with breast tumors.

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arXiv2022

Directional testing for high-dimensional multivariate normal distributions

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 under the null hypothesis the directional test outperforms not only the usual first and higher-order finite-$p$ solutions but also alternative methods tailored for high-dimensional settings. Simulation results also indicate that the power performance of the different tests depends on the specific alternative hypothesis.

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