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Asymptotic expansions in mean and covariance structure analysis

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  • Ogasawara, Haruhiko

Abstract

Asymptotic expansions of the distributions of parameter estimators in mean and covariance structures are derived. The parameters may be common to, or specific in means and covariances of observable variables. The means are possibly structured by the common/specific parameters. First, the distributions of the parameter estimators standardized by the population asymptotic standard errors are expanded using the single- and the two-term Edgeworth expansions. In practice, the pivotal statistic or the Studentized estimator with the asymptotically distribution-free standard error is of interest. An asymptotic distribution of the pivotal statistic is also derived by the Cornish-Fisher expansion. Simulations are performed for a factor analysis model with nonzero factor means to see the accuracy of the asymptotic expansions in finite samples.

Suggested Citation

  • Ogasawara, Haruhiko, 2009. "Asymptotic expansions in mean and covariance structure analysis," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 902-912, May.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:902-912
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    References listed on IDEAS

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    1. Ogasawara, Haruhiko, 2007. "Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1726-1750, October.
    2. Ogasawara, Haruhiko, 2006. "Asymptotic expansion of the sample correlation coefficient under nonnormality," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 891-910, February.
    3. Yuan, Ke-Hai & Hayashi, Kentaro & Bentler, Peter M., 2007. "Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1262-1282, July.
    4. Haruhiko Ogasawara, 2004. "Asymptotic biases in exploratory factor analysis and structural equation modeling," Psychometrika, Springer;The Psychometric Society, vol. 69(2), pages 235-256, June.
    5. Ogasawara, Haruhiko, 2007. "Higher-order Estimation Error in Structural Equation Modeling," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 57(4), pages 131-160.
    6. Ke-Hai Yuan & Peter Bentler, 2006. "Mean Comparison: Manifest Variable Versus Latent Variable," Psychometrika, Springer;The Psychometric Society, vol. 71(1), pages 139-159, March.
    7. Haruhiko Ogasawara, 2001. "Standard errors of fit indices using residuals in structural equation modeling," Psychometrika, Springer;The Psychometric Society, vol. 66(3), pages 421-436, September.
    8. Ogasawara, Haruhiko, 2008. "Errata and supplement to the paper "Higher-order asymptotic cumulants of Studentized estimators in covariance structures"," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 59(2/3), pages 95-107.
    9. Boik, Robert J., 2008. "An implicit function approach to constrained optimization with applications to asymptotic expansions," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 465-489, March.
    10. Yiu-Fai Yung & Peter M. Bentler, 1999. "On Added Information for ML Factor Analysis with Mean and Covariance Structures," Journal of Educational and Behavioral Statistics, , vol. 24(1), pages 1-20, March.
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    Cited by:

    1. Shimizu, Hiroaki & Wakaki, Hirofumi, 2011. "Asymptotic expansions for a class of tests for a general covariance structure under a local alternative," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1080-1089, July.
    2. Ogasawara, Haruhiko, 2009. "Supplement to Ogasawara's papers on "the ADF pivotal statistic", "mean and covariance structure analysis", and "maximal reliability"," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 60(1), pages 21-44.
    3. Ogasawara, Haruhiko, 2017. "Expected predictive least squares for model selection in covariance structures," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 151-164.
    4. Ogasawara, Haruhiko, 2016. "Bias correction of the Akaike information criterion in factor analysis," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 144-159.
    5. Ogasawara, Haruhiko, 2012. "Cornish-Fisher expansions using sample cumulants and monotonic transformations," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 1-18, January.

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