DEVELOPMENT OF SOME INFLUENTIAL MEASURES IN ONE PARAMETER NEW RIDGE-TYPE ESTIMATORS OF LINEAR REGRESSION MODELS

Abstract

In linear regression analysis, influential points are observations that exert disproportionate impact on regression models. If not properly identified and addressed, they can distort parameter estimates, fitted values, and statistical inferences. Although, Ordinary Least Squares (OLS) estimator is widely used for estimating parameter coefficients in regression analysis, it performs poorly when multicollinearity and outliers are present. Many existing biased estimators are developed to mitigate only multicollinearity, while robust regressions are used to reduce the effect of outliers on regression estimates. In practice, these two problems often occur together. Similarly, traditional influential diagnostic measures developed under the OLS framework, such as Cook’s Distance (CD) and Difference in Fits (DFFITS), may not perform well when multicollinearity and outliers exist simultaneously. Hence, this study developed influential diagnostic measures within One-parameter new ridge-type estimator’s framework, to improve detection of influential points when multicollinearity and outliers co-exist in a dataset. The Kibria–Lukman (KL) estimator was combined with six robust regression techniques, namely: Modified Maximum Likelihood (MM), Least Absolute Deviation (LAD), Least Median of Squares (LMS), Least Quartile of Squares (LQS), Least Trimmed Squares (LTS), and S estimators to produce a class of Robust Kibria–Lukman (RKL) estimators. Based on these estimators, generalized forms of Cook’s Distance, DFFITS, Difference in Betas (DFBETAS), Covariance Ratio’s, Hadi’s, Pena’s, and Atkinson’s statistics were derived, along with their approximate case-deletion expressions for Cook’s Distance and DFFITS. Their performance were evaluated through a Monte Carlo simulation study, replicated 1,000 times, with sample sizes from 10 to 500, correlation levels from 0.80 to 0.99, outlier magnitudes from 0 to 10, and contamination rates of 10% and 20%. The findings showed that the Covariance Ratio based on the Robust Kibria–Lukman_M estimator (CVR_KL_M) was the overall best-performing diagnostic measure, with the highest total frequency of 100% detection (2,622). It was followed by CVR_KL_LAD (2,603) and CVR_KL_MM (2,325). CVR_KL_M also showed stable performance across different sample sizes, with detection frequencies of 314 at n=10, 330 at n=20, 30, 50, 100 and 250 and 329 at n=40 and 500. In real life data applications, the proposed measures identified observations 2, 16, 1, 12, and 9 as influential in the economic dataset, while several existing methods failed to detect observations 2, 12, and 9. In the Hussein dataset, the proposed methods consistently identified observations 15, 20, 16, 12, and 4 as influential. The study concluded that the proposed Robust Kibria–Lukman influential measures were more effective, stable, and reliable than the existing diagnostic measures for detecting influential points in a dataset affected by both multicollinearity and outliers. Particularly, CVR_KL_M emerged as the most efficient diagnostic measure. The study therefore provides a statistically sound and practically useful approach for identifying influential points in linear regression models under these combined conditions. It is therefore recommended that applied researchers and data analysts should use the proposed Robust Kibria–Lukman-based diagnostic measures, especially CVR_KL_M, when analysing regression data affected by both multicollinearity and outliers.