| Summary: | A function $f(\underline{x})$ which increases each time we transpose an out of order pair of coordinates, $x\sb{j} > x\sb{k}$ for some $j x\sb{k}$ by transposing the two x coordinates. The theory of AI functions is tailor made for ranking and selection problems, in which case we assume that the density $f(\underline{\theta}$,$\underline{x})$ of observations with respective parameters $\theta\sb1, \..., \theta\sb{n}$ is AI, and the goal is to determine the largest or smallest parameters. === In this dissertation we present new applications of AI functions in such areas as biology and reliability, and we generalize the notion of AI functions. We consider multivector extensions, some with and one without respect to parameter vectors, and we connect these. Another generalization (TEGO) is motivated by the connection between total positivity (TP) and AI. TEGO results are shown to imply AI and TP results. We also define and develop a partial ordering on densities of rank vectors. The theory, which involves finding the extreme points of the convex set of AI rank densities, is then used to establish some power results of rank tests. === Source: Dissertation Abstracts International, Volume: 50-08, Section: B, page: 3563. === Major Professor: Fred Leysieffer. === Thesis (Ph.D.)--The Florida State University, 1989.
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