The classical F statistic for testing parameter vector in linear models is well-known to be robust within the class of spherical distributions. The validity of this robustness follows from the uniqueness of uniform distribution on the unit sphere which, like the normal distribution, belongs to the spherical class. The statistic, however, can not be used (not even under normality) when the data are high-dimensional, i.e., when the dimension of the parameter vector exceeds the sample size. This article introduces a modified version of the F statistic, and demonstrates that the robustness of the classical statistic extends to the modified version for high-dimensional case. In particular, the link between the normal distribution and the uniform distribution on the unit sphere (now in high-dimensional spaces) is again the key tool. Since both distributions are sub-Gaussian, the result overlaps with the sub-Gaussian class as well. In fact, under the bounded fourth moment assumption, the result is shown to extend much more generally. Extension to the linear mixed models is also given. From practical standpoint, the proposed modification is very simple to apply, covers a wide variety of models, and is valid under a few mild assumptions. Simulations are used to show the accuracy of the proposed theory. Applications on several real data sets are demonstrated.
Rauf Ahmad (Mon,) studied this question.