Linear hypothesis testing in high-dimensional one-way MANOVA

Abstract

In recent years, with the rapid development of data collecting technologies, high-dimensional data have become increasingly prevalent. Much work has been done for testing hypotheses on mean vectors, especially for high-dimensional two-sample problems. Rather than considering a specific problem, we are interested in a general linear hypothesis testing (GLHT) problem on mean vectors of several populations, which includes many existing hypotheses about mean vectors as special cases. A few existing methodologies on this important GLHT problem impose strong assumptions on the underlying covariance matrix so that the null distributions of the associated test statistics are asymptotically normal. In this paper, we propose a simple and adaptive test based on the L 2 -norm for the GLHT problem. For normal data, we show that the null distribution of our test statistic is the same as that of a chi-squared type mixture which is generally skewed. Therefore, it may yield misleading results if we blindly approximate the underlying null distribution of our test statistic using a normal distribution. In fact, we show that the null distribution of our test statistic is asymptotically normal only when a necessary and sufficient condition on the underlying covariance matrix is satisfied. This condition, however, is not always satisfied and it is not an easy task to check if it is satisfied in practice. To overcome this difficulty, we propose to approximate the null distribution of our test statistic using the well-known Welch–Satterthwaite chi-squared approximation so that our new test is applicable without any assumption on the underlying covariance matrix. Simple ratio-consistent estimators of the unknown parameters are obtained. The asymptotic and approximate powers of our new test are also investigated. The methodologies are then extended for non-normal data. Four simulation studies and a real data application are presented to demonstrate the good performance of our new test compared with some existing testing procedures available in the literature.