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Majorization: its extensions and the preservation theorems

University of British Columbia

This thesis deals mainly with the orderings induced by majorizatIon, the two weak majorizations and their associated inequalities. One of these weak majorizations has received some attention in the literature. However, the other one, being dual to the former, is totally overlooked. Schur functions which preserve these orderings are shown to have applications in statistics. In Chapter 1, we discuss briefly the majorizations and their relations with Schur functions, bringing out the extensions to the weak majorizations where possible. In Chapter 2, we generalize the ordinary majorizations to those parametrized by a vector p of positive components. We discuss the properties of these new majorizations in a direction parallel to that of ordinary majorizations. The Schur functions are likewise generalized. In Chapter 3, we discuss the stochastic extensions of majorizations and the preservation theorem of Schur convexity due to Proschan and Sethuraman (1977). With a preservation theorem on monotonicity, we study the stochastic extensions of the weak majorizations. Some inequalities arising in some multivariate distributions are found to be direct consequences of the two preservation theorems. Finally, we consider the unbiasedness of a certain class of tests of significance.

Text

2010-02-16

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http://hdl.handle.net/2429/20265

Mathematics

University of British Columbia

Graduate