Multivariate Statistical Methods for Nonnormal Populations

人们对多变量分析的理论和应用进行了深入的研究。多变量正态分布一直是关注的焦点。这主要是由于其他分布的数学难以处理，并且开发的许多程序在正常模型中显示出最优性质。然而，在实践中，经常会出现正态性假设不成立的情况。本研究的目的是探讨多元非正态模型的统计理论和方法。以下是通过研究项目获得的结果。在正态和非正态模型中检验了线性判别分析中选择变量的各种程序的性能。基于Edgeworth展开式中对正态分布的收敛速度，构造了一种通用的归一化原理。对推导高阶埃奇沃斯展开式时出现的问题进行了研究。研究了非参数情况下多变量分析中参数置信区间的构造问题。给出了基于估计量的归一化变换的处理方法。考虑了Bootstrap置信区间与我们的程序之间的关系。得到了非正态模型的变异系数分布和正态变量的二次型分布的高阶渐近展开式。使用计算机代数系统REDUCE-III.5克服了计算的困难。对于正态分布或其他分布的尺度混合，导出了渐近展开式。得到了它们的误差范围。基于经验分布函数的鞅项构造了拟合优度检验的统计量，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.