M. Rauf Ahmad: Some Tests for High Dimensional Multivariate Data

In this talk, some tests for the mean vector and the covariance matrix are presented when the dimension of the multivariate vector, p, may exceed the sample size, n, i.e., the large-p small-n problem. Although, the tests are basically developed under the assumption of multivariate normality, they are also shown to be robust when normality assumption is replaced with certain mild assumptions on the covariance matrix. However, nowhere is the condition on the relationship between n and p needed. For example, it is not required that, when p gets very large, the ratio p/n is bounded away from 0 and infinity. Another important feature of the tests is the use of the theory of U-statistics to derive asymptotic distributions of some of these tests. The test statistics are shown to be asymptotically normally distributed and the accuracy of the statistics is not disturbed even if p >> n.