The Annals of Statistics

The Asymptotic Normal Distribution of Estimators in Factor Analysis under General Conditions

T. W. Anderson and Yasuo Amemiya

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Asymptotic properties of estimators for the confirmatory factor analysis model are discussed. The model is identified by restrictions on the elements of the factor loading matrix; the number of restrictions may exceed that required for identification. It is shown that a particular centering of the maximum likelihood estimator derived under assumed normality of observations yields an asymptotic normal distribution that is common to a wide class of distributions of the factor vectors and error vectors. In particular, the asymptotic covariance matrix of the factor loading estimator derived under the normal assumption is shown to be valid for the factor vectors containing a fixed part and a random part with any distribution having finite second moments and for the error vectors consisting of independent components with any distributions having finite second moments. Thus the asymptotic standard errors of the factor loading estimators computed by standard computer packages are valid for virtually any type of nonnormal factor analysis. The results are extended to certain structural equation models.

Article information

Ann. Statist., Volume 16, Number 2 (1988), 759-771.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62E20: Asymptotic distribution theory

Factor analysis asymptotic normality asymptotic covariance matrix linear functional relationship structural equation model


Anderson, T. W.; Amemiya, Yasuo. The Asymptotic Normal Distribution of Estimators in Factor Analysis under General Conditions. Ann. Statist. 16 (1988), no. 2, 759--771. doi:10.1214/aos/1176350834.

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