Asymptotic Expansion and Conditional Robustness for the Sample Multiple Correlation Coefficient Under Nonnormality

ABSTRACT

Asymptotic distributions of the standardized estimators of the squared and non squared multiple correlation coefficients under nonnormality were obtained using Edgeworth expansion up to O(1/n). Conditions for the normal-theory asymptotic biases and variances to hold under nonnormality were derived with respect to the parameter values and the weighted sum of the cumulants of associated variables. The condition for the cumulants indicates a compensatory effect to yield the robust normal-theory lower-order cumulants. Simulations were performed to see the usefulness of the formulas of the asymptotic expansions using the model with the asymptotic robustness under nonnormality, which showed that the approximations by Edgeworth expansions were satisfactory.

Keywords:

• Asymptotic bias
• Asymptotic robustness
• Cumulants
• Edgeworth expansion
• Multiple correlation coefficient
• Nonnormality

Mathematics Subject Classification:

• 62J05
• 62E20

Acknowledgments

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

Notes

e.g. Note. n = Sample size minus 1 in the simulation; Th.=Theoretical or asymptotic values; Nm, T9, C10, and C3=Normal, t- (df = 9), and chi-square (df = 10, 3) distributions, respectively; C2U = Mixture of chi-square (df = 2) and uniform distributions; SD = Standard deviation from simulation; HASE, ASE; The asterisks indicate that the corresponding normal-theory values hold.

e.g. Note. n = Sample size minus 1 in the simulation; Th.=Theoretical or asymptotic values; Nm, T9, C10, and C3=Normal, t- (df = 9), and chi-square (df = 10, 3) distributions, respectively; C2U = Mixture of chi-square (df = 2) and uniform distributions; SD = Standard deviation from simulation; HASE, ASE. The asterisks indicate that the corresponding normal-theory values hold.

e.g. Note. n = Sample size minus 1 in the simulation; Th. = Theoretical or asymptotic values; Nm, T9, C10, and C3=Normal, t- (df = 9), and chi-square (df = 10, 3) distributions, respectively; C2U=Mixture of chi-square (df = 2) and uniform distributions; SD = Standard deviation from simulation; HASE, ASE. The asterisks indicate that the corresponding normal-theory values hold.

Note. Sim.=Simulated values; Th.=Theoretical or asymptotic values; Nm, T9, C10 and C3=Normal, t- (df = 9), and chi-square (df = 10, 3) distributions, respectively; C2U=Mixture of chi-square (df = 2) and uniform distributions; SD=Standard deviation from simulation; HASE, ASE.