Multivariate Statistical Methods for Nonnormal Populations

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Intensive investigations have been made concerning the theory and applications of multivariate analysis. Attension has been focused to mustivariate normal distribution.This is mainly due to the fact that the mathematics is intractable for other distributions and many of the procedures developed are shown to have optimal properties in normal model.In practice,However,there often occur the cases in which the normality assumption does not hold.The purpose of this research is to investigate statistical theory and methods in multivariate nonnormal models.The followings are results obtained through the research project.1.The performance of various procedures for selecting variables in linear discriminant analysis was examined both in normal and in nonnormal models.2.A general principle of normalization was constructed based on the rate of convergence to the normal distribution in an Edgeworth expansion.Investigation was made in connection with the problem which arises in deriving higher order Edgeworth expansions.3.The problem of constructing confidence intervals for parameters in multivariate analysis was considered in nonparametric situations.Some procedures were given based on normalizing transformations of estimators. The relationship between Bootstrap confidence intervals and our procedures was considered.4.Higher order asymptotic expansions were obtained for the distributions of the coeffient of variation in nonnormal models and of quadratic forms in normal variables.The hardness of computation was overcome by the use of a computer algebra system,REDUCE-III.5.Asymptotic expansions were derived for the scale mixtures of the normal or of other distributions.Their error bounds were also obtained.6.Some statistics for goodness-of fit tests were constructed based on the martingale term of the empirical distribution function and examined their properties.