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

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(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.