In this dissertation we assume that the observations are from normal populations but are correlated and study the problem of characterizing the class of covariance structures such that the distributions of the popular test statistics remain invariant, that is, they remain the same except for a constant factor. We first obtain some simple extensions and variations of the well known Cauchy-Schwarz inequality. Incidentally, several inequalities that are useful in the detection of outliers can be deduced from our results.
Our main result is a characterization of the class of all nonnegative definite solutions W to the matrix equation AWA = B, where A is a symmetric and B is a nonnegative definite matrix. We illustrate the proof of this characterization by considering a special case where A = B = A* = I - 1/n ee\u27, I is the identity matrix and e is a vector of ones. We thus have an elegant characterization of the class of all nonnegative definite g-inverses of the centering matrix A*. Next we present the statistical applications of our matrix theoretic results. For example, we show that the usual two sample t-statistic has a t-distribution if the observations in one of the samples are positively equicorrelated and those in the other sample are negatively equicorrelated with the same correlation in absolute value. More generally, we have a complete characterization of the class of covariance matrices for which the distributional properties of the quadratic forms in ANOVA problems remain invariant. These results are contained in Chapter 3.
In Chapters 4 and 5, we generalize our results to the multivariate test statistics, first considering a special covariance structure that occurs in repeated measurements and later for an arbitrary covariance structure. These include invariance properties of the distributions of quadratic forms in MANOVA problems and one- and two-sample Hotelling\u27s T2 statistics. As preliminaries to the multivariate results, we obtain a very general version of the Cochran\u27s theorem concerning the independence and Wishartness of the multivariate quadratic forms
