Higher-Order Asymptotic Standard Error and Asymptotic Expansion in Principal Component Analysis

ABSTRACT

Asymptotic expansions of the distributions of the estimators of unrotated and orthogonally rotated component loadings are given under non normality of observed variables in principal component analysis for sample covariance and correlation matrices. The expansions include those for the Studentized statistics of the estimators with unknown standard errors. The expansions with the adjustment of the higher-order asymptotic variance of estimators are also presented with weight for partial adjustment. The formula is applied to the estimators of the contributions of unrotated/rotated components as well as their loadings, which includes eigenvalues as special cases. Simulations were performed to see the accuracy of the asymptotic moments and the higher-order standard errors in samples with finite sample sizes.

Keywords:

• Bias
• Edgeworth expansion
• Higher-order asymptotic variances
• Principal component analysis
• Skewness
• Varimax rotation

Mathematics Subject Classification:

• 62H25
• 62E20

Acknowledgment

Partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology (C) (2) #16500167.

Notes

Note. I (II) = Loadings of Component I (II), Ct. I (II) = Contribution of Component I (II), SD-t = Standard deviations of Studentized parameter estimates from simulation, Th. = Theoretical (asymptotic) values, Sim. = Simulated values.

Note. I (II) = Loadings of Component I (II), Ct. I (II) = Contribution of Component I (II), SD-t = Standard deviations of Studentized parameter estimates from simulation, Th. = Theoretical (asymptotic) values, Sim. = Simulated values.

Note. I (II) = Loadings of Component I (II), Ct. I (II) = Contribution of Component I (II), SE = Asymptotic standard errors of order n ^−1/2, HSE = Higher-order asymptotic standard errors up to order n ⁻¹ with bias-adjustment, SD = Standard deviations from simulation.

Note. I (II) = Loadings of Component I (II), Ct. I (II) = Contribution of Component I (II), SE = Asymptotic standard errors of order n ^−1/2, HSE = Higher-order asymptotic standard errors up to order n ⁻¹ with bias-adjustment, SD = Standard deviations from simulation.

Note: The decimal points are omitted (actual values are to be multiplied by 10⁻⁵).

I (II) = Loadings of Component I (II), Ct. I (II) = Contribution of Component I (II),

E. = Edgeworth expansion.

Note: The decimal points are omitted (actual values are to be multiplied by 10⁻⁵).

I (II) = Loadings of Component I (II), Ct. I (II) = Contribution of Component I (II),

E. = Edgeworth expansion.