Multivariate Statistical Methods for Nonnormal Populations

人们对多元分析的理论和应用进行了深入的研究。多元正态分布一直是人们关注的焦点，这主要是由于数学对于其他分布是难以处理的，并且许多已开发的程序在正态模型中被证明具有最优性质，然而，在实践中，在多元非正态模型中，经常会出现正态性假设不成立的情况，本研究旨在探讨多元非正态模型的统计理论和方法，本文主要研究了正态和非正态模型下线性判别分析中变量选择的各种方法的性能。2.根据Edgeworth展开式向正态分布收敛的速度，建立了一般的归一化原理，并对高阶Edgeworth展开式的推导问题进行了研究。3.对正态模型和非正态模型进行了线性判别分析，得到了一些新的结果在非参数情形下，考虑了多元分析中参数置信区间的构造问题，给出了基于估计量归一化变换的构造方法。4.得到了非正态模型变差系数分布和正态变量二次型分布的高阶渐近展开式，并利用计算机代数系统克服了计算困难，REDUCE-III.5.导出了正态分布或其它分布的尺度混合的渐近展开式，并得到了它们的误差界。基于经验分布函数的鞅项构造了拟合检验，并检验了它们的性质。

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.