In this dissertation, a new test statistic for testing for a block diagonal covariance matrix structure with a multivariate normal population where the number of variables p exceeds the number of observations n is proposed. Whereas classical approaches such as the likelihood ratio test cannot be applied when p n  , the proposed test statistic is based on the ratio of the estimators of 2 tr and 2 trD , where  is the population covariance matrix and D is the population covariance matrix under the null hypothesis. Furthermore, ...