A novel scale-space approach for multinormality testing and the k-sample problem in the high dimension low sample size scenario

Abstract

Two classical multivariate statistical problems, testing of multivariate normality and the <i>k</i>-sample problem, are explored by a novel analysis on several resolutions simultaneously. The presented methods do not invert any estimated covariance matrix. Thereby, the methods work in the High Dimension Low Sample Size situation, i.e. when <i>n</i> ≤ <i>p</i>. The output, a significance map, is produced by doing a one-dimensional test for all possible resolution/position pairs. The significance map shows for which resolution/position pairs the null hypothesis is rejected. For the testing of multinormality, the Anderson-Darling test is utilized to detect potential departures from multinormality at different combinations of resolutions and positions. In the <i>k</i>-sample case, it is tested whether <i>k</i> data sets can be said to originate from the same unspecified discrete or continuous multivariate distribution. This is done by testing the <i>k</i> vectors corresponding to the same resolution/position pair of the k different data sets through the <i>k</i>-sample Anderson-Darling test. Successful demonstrations of the new methodology on artificial and real data sets are presented, and a feature selection scheme is demonstrated.