Performance evaluation of discrepancy function estimators in the presence of outliers

Abstract

Small sample size or large number of manifest variables had been shown to affect estimators of structural equation modelling (SEM). This poses problems in a form of matrix singularity, non-convergence, unreliable results and rejection of good models. Although some estimators do better than others, all of them do better with increase in sample size, especially asymptotic distribution free estimator (ADFE). Yuan and Chan developed a ridge maximum likelihood estimator (RMLEa) which performs better than the maximum likelihood estimator (MLE) when sample sizes are small by using a modified covariance matrix where a constant (a), computed by dividing the number of manifest variables (k) by sample size (N), is added to the main diagonal of the sample covariance matrix (S). The RMLEa reported consistent estimates which also has better efficiency than the MLE but the constant lacks much information from the data. In this study, a modified ridge maximum likelihood estimator (RMLEh) is proposed to include more information from the data and compared with the RMLEa, MLE, generalized least squares (GLSE), unweighted least squares (ULSE) and ADFE in terms of estimating mean and variance of the discrepancy function in the presence of different levels of positive extreme outliers and sample sizes. The convergence rate, relative mean square error, relative bias, coefficient of variation and run-time were used as performance indicators. The results show that, for small sample sizes (N = 10 to 50), the maximum likelihood family of estimators reported the highest convergence rate and more quality mean and variance discrepancy function than others with RMLEh performing better than RMLEa and MLE. The maximum likelihood family of estimators provide a more stable and reliable estimates when data are perturbed. As expected, the ADFE did well with increase in sample sizes but was unstable with perturbed data. Mostly, the ULSE did poorly but had a better convergence rate than the GLSE for small samples.