Invariance Properties of Statistical Tests for Dependent Observations

<p>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.</p> <p>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', 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.</p> <p>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's T² statistics. As preliminaries to the multivariate results, we obtain a very general version of the Cochran's theorem concerning the independence and Wishartness of the multivariate quadratic forms.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics & Statistics

Year dc:date.available

1994

Author dc:creator

• Vaish, Akhil K.

Contributors dc:contributor

• N. R. Chaganty
• R. C. Dahiya
• D. N. Naik
• E. P. Markowski

• Covariance matrices
• Matrix theory
• Nonnegative definite solutions
• Test statistics
• Statistics and Probability

Statement dc:rights

• <p>In Copyright. URI: <a href="http://rightsstatements.org/vocab/InC/1.0/">http://rightsstatements.org/vocab/InC/1.0/</a> This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).</p>

Repository record dc:identifier

https://digitalcommons.odu.edu/mathstat_etds/71

OAI identifier oai:identifier

oai:digitalcommons.odu.edu:mathstat_etds-1067