UBC Theses and Dissertations
Multiple comparison methods and certain distributions arising in multivariate statistical analysis Bagai, Om Parkash
The problem of classifying multivariate normal populations into homogeneous clusters on the basis of random samples drawn from those populations is taken up. Three alternative methods have been suggested for this. One of them is explained fully with an illustrative example, and the tabular values for the corresponding statistic, used for the purpose, have been computed. In the case of the other two alternatives only the working procedure is discussed. Further, a new statistic R, 'the largest distance', is proposed in one of these two alternatives, and its distribution is determined for the bivariate case in the form of definite integrals. Ignoring a priori probabilities, two alternative methods are suggested for assigning an arbitrary population to one or more clusters of populations, and are demonstrated by an illustrative example. A method is discussed for finding confidence regions for the non-centrality parameters of the distributions of certain statistics used in multivariate analysis and this method is also illustrated by an example. The exact distribution of the determinant of the sum of products (S.P.) matrix is found (in series), both in the central and the non-central linear cases for particular values of the rank of the matrix. Further, these results have been made use of in finding the limiting distribution of the Wilks-Lawley statistic proposed for testing the null hypothesis of the equality of the mean vectors of any number of populations. Six different statistics based on the roots of certain determinantal equations have been proposed for various tests of hypotheses arising in the problems of multivariate analysis of variance (Anova). Their distributions in the limited cases of two and three eigenroots have been found in the form of definite integrals. Also, the limiting distribution of the Roy's statistics of the largest, an intermediate and the smallest eigenroots have been found by a simple, easy method of integration, which method is quite different from that of Nanda (1948). Lastly, the distributions of the mean square and the mean product (M.P.) matrix have been approximated respectively in the univariate and multivariate cases of unequal sub-class numbers in the analysis of variance (Anova) of Model II.
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