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

(1)渐近展开方法Edgeworth展开及其变化已被用作基本工具。由于数据在实践中或多或少是非正态分布，任意分布，包括正常的一个作为特殊情况下，假设为可观察变量。(2)利用Edgeworth展开及相关方法，得到了非正态分布下样本相关系数和样本多重相关系数的1/n阶渐近展开式。研究了正态理论低阶渐近累积量的渐近鲁棒性。(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。对于Studentization，需要达到四阶的样本累积量，并且期望在有限样本中是不稳定的。考虑到这一性质，给出了正态和非正态下的Studentized估计。(4)样本系数α的渐近展开样本系数α在行为科学中被用作可靠性指标。可靠性指标定义为相关矩阵以及协方差矩阵。估计量的分布的展开式已使用Edgeworth展开式导出。

(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.