The Infinitesimal Jackknife and Moment Structure Analysis Using Higher Order Moments

Area of Science:

• Statistics
• Econometrics
• Psychometrics

Background:

• Higher-order sample moments are asymptotically normally distributed.
• Current literature and software provide incorrect estimates for their asymptotic covariance matrices.
• Accurate covariance matrix estimation is vital for statistical inference and model evaluation.

Purpose of the Study:

• To introduce and demonstrate the infinitesimal jackknife as a correct method for estimating asymptotic covariance matrices of higher-order sample moments.
• To highlight the advantages of the infinitesimal jackknife, including its ease of use with stacked or subsetted estimators.
• To apply these corrected estimates for testing the goodness of fit in non-linear factor analysis models.

Main Methods:

• Introduction to the infinitesimal jackknife (IJ) method.
• Application of IJ to correctly estimate asymptotic covariance matrices of higher-order sample moments.
• Development of a computationally accelerated form for IJ estimates.

Main Results:

• Demonstrated that existing methods for estimating asymptotic covariance matrices of higher-order sample moments are incorrect.
• Showcased the infinitesimal jackknife as a valid and advantageous approach for accurate estimation.
• Successfully utilized IJ estimates to test the goodness of fit for a non-linear factor analysis model.

Conclusions:

• The infinitesimal jackknife provides a correct and flexible method for estimating asymptotic covariance matrices of higher-order sample moments.
• This method enhances the reliability of statistical inference and model testing, particularly in complex models like non-linear factor analysis.
• The computationally accelerated form of IJ improves practical applicability.