PARTIAL ORDERINGS, WITH APPLICATIONS TO RELIABILITY (PARTIAL ORDERINGS, SCHUR-OSTROWSKI THEOREM, INEQUALITIES)

• Other Authors: CHAN, WAI TAT.
• Format: Others
• Subjects: Statistics
• Online Access: http://purl.flvc.org/fsu/lib/digcoll/etd/3086034

This dissertation is a contribution to the use of inequalities in reliability theory. Specifically, we study three partial orderings, develop some useful properties of these orderings, and apply them to obtain several applications in reliability. === The first partial ordering is the notion of convex-ordering among life distributions. This is in the spirit of Hardy, Littlewood, and Polya (1952) who introduced the concept of relative convexity. Many parametric families of distribution functions encountered in reliability theory are convex-ordered. Different coherent structures can also be compared with respect to this partial ordering. === The second partial ordering is the ordering of majorization among integrable functions. This ordering is a generalization of the majorization ordering of Hardy, Littlewood, and Polya (1952) for vectors in n-dimensional Euclidean spaces. The concept of majorization among vectors plays a fundamental role in establishing various inequalities. These inequalities can be recast as statements that certain functions are increasing with respect to the ordering of majorization. Such functions are called Schur-convex functions. An important result in the theory of majorization is the Schur-Ostrowski Theorem, which characterizes Schur-convex functions. A functional defined on the space of integrable functions is said to be Schur-convex if it is increasing with respect to the ordering of majorization. We obtain an analogue of the Schur-Ostrowski theorem which characterizes Schur-convex functionals in terms of their Gateaux differentials. === The third partial ordering is the ordering of unrestricted majorization among integrable functions. This partial ordering is similar to majorization but does not involve the use of decreasing rearrangements. We establish another analogue of the Schur-Ostrowski Theorem for functionals increasing with respect to the partial ordering of unrestricted majorization. === Source: Dissertation Abstracts International, Volume: 46-03, Section: B, page: 0891. === Thesis (Ph.D.)--The Florida State University, 1985.