# The Infinitesimal Jackknife and Moment Structure Analysis Using Higher Order Moments

## Abstract

Mean corrected higher order sample moments are asymptotically normally distributed. It is shown that both in the literature and popular software the estimates of their asymptotic covariance matrices are incorrect. An introduction to the infinitesimal jackknife is given and it is shown how to use it to correctly estimate the asymptotic covariance matrices of higher order sample moments. Another advantage in using the infinitesimal jackknife is the ease with which it may be used when stacking or sub-setting estimators. The estimates given are used to test the goodness of fit of a non-linear factor analysis model. A computationally accelerated form for infinitesimal jackknife estimates is given.

## Keywords

Kronecker products Asymptotically normal Influence functions Pseudo-values Goodness of fit Acceleration## Notes

### Acknowledgments

The authors would like to thank the review team and in particular reviewer 2 who motivated the inclusion of the offline material. The research of the second author is supported by Grant EC02011-28875 from the Spanish Ministry of Science and Innovation.

## Supplementary material

## References

- Bentler, P. M. (2012).
*EQS 6 structural equations program manual*. Encino, CA: Multivariate Software.Google Scholar - Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures.
*British Journal of Mathematical and Statistical Psychology*,*37*, 62–83.Google Scholar - Cragg, J. G. (1997). Using higher order moments to estimate the simple errors-in-variables model.
*The Rand Journal of Economics*, S71–S91. Special Issue in honor of Richard E. Quandt.Google Scholar - Cramer, H. (1946).
*Mathematical methods of statistics*. Princeton, NJ: Princeton University Press.Google Scholar - Efron, B., & Tibshirani, R. J. (1993).
*An Introduction to the Bootstrap*. New York, NY: Chapman & Hall.CrossRefGoogle Scholar - Hausman, J. A., Newey, W. K., Ichimura, H., & Powell, J. L. (1991). Identification and estimation of polynomial errors-in-variables models.
*Journal of Econometrics*,*50*, 273–295.CrossRefGoogle Scholar - Jaeckel, L. (1972).
*The infinitesimal jackknife, Memorandum #MM 72–1215-11*. Murray Hill, NJ: Bell Laboratories.Google Scholar - Jennrich, R. I. (2008). Nonparametric estimation of standard errors in covariance structure analysis.
*Psychometrika*,*73*, 579–594.Google Scholar - Jennrich, R. I., & Satorra, A. (2013). Continuous orthogonal complement functions and distribution-free tests in moment structure analysis.
*Psychometrika*,*78*, 545–552.CrossRefPubMedGoogle Scholar - Mooijaart, A. (1985). Factor analysis of non-normal variables.
*Psychometrika*,*50*, 323–342.CrossRefGoogle Scholar - Mooijaart, A., & Bentler, P. M. (2010). An alternative approach for non-linear latent variable models.
*Structural Equation Modeling*,*17*, 357–373.CrossRefGoogle Scholar - Neyman, J. (1937). Remarks on a paper by E. C. Rhodes.
*Journal of the Royal Statistical Society*,*100*, 50–57.Google Scholar - Ozaki, K., Toyoda, H., Iwama, N., Kubo, S., & Ando, J. (2011). Using non-normal SEM to resolve the ACDE model in the classical twin design.
*Behavioral Genetics*,*41*, 329–339.CrossRefGoogle Scholar