Asymptotic expansions of the estimators in covariance structures with some robustness issues on normal-theory asymptotic cumulants under nonnormality

(1)渐近展开法Edgeworth展开式及其变分被用作基本工具。由于实际数据或多或少都是非正态分布的，所以对于可观测变量假定为任意分布，包括正态分布作为特例。(2)样本相关系数的渐近展开式利用Edgeworth展开式及相关方法，得到了非正态下样本相关系数和样本多重相关系数的渐近展开式，可达1/n阶。研究了正态理论低阶渐近累积量的渐近稳健性。(3)因子分析中的渐近展开给出了因子分析中非学生化估计和学生化估计的渐近展开式。对于学化化，需要高达四阶的样本累积量，并且预计在有限样本中是不稳定的。考虑到这一性质，给出了正态下和非正态下的学生化估计。(4)样本系数α的渐近展开在行为科学中，样本系数α被用作可靠性指标。定义了相关矩阵和协方差矩阵的可靠性指标。利用Edgeworth展开式得到了估计量分布的展开式。

(1) Methods of asymptotic expansionEdgeworth expansion and its variations have been employed as basic tools. Since data in practice are more or less nonnormally distributed, arbitrary distributions including the normal one as a special case are assumed for observable variables.(2) Asymptotic expansions for the sample correlation coefficientAsymptotic expansions for the sample correlation coefficient and the sample multiple correlation coefficient are obtained up to order 1/n under nonnormality using the Edgeworth expansion and related methods. Asymptotic robustness of the normal-theory lower-order asymptotic cumulants are also investigated.(3) Asymptotic expansions in factor analysisAsymptotic expansions of the non-Studentized estimators and the Studentized ones in factor analysis are given. For Studentization, the sample cumulants up to the fourth order are required and expected to be unstable in finite samples. Considering this property, the Studentized estimators are given under normality and under nonnormality.(4) Asymptotic expansion for the sample coefficient alphaThe sample coefficient alpha is used as an index of reliability in the behavioral sciences. Reliability indexes are defined for correlation matrices as well as covariance matrices. The expansions of the distributions of the estimators have been derived using the Edgeworth expansion.