Influence Analysis Using Liu–Ridge Estimators in Semiparametric Regression Models

Influential observations and the multicollinearity among predictors in semiparametric regression models (SPRMs) often lead to biased and unstable coefficient estimates, as well as distorted patterns in leverage and residuals. To solve these problems, we propose two-parameter Liu–Ridge penalized least squares estimators (TLRPLSEs) and derive an approximate case-deletion formula combining diagnostic measures for influential observations in SPRMs. Cook’s distance, along with a case-deletion approach, is employed to evaluate their performance. The effectiveness of the proposed estimators was evaluated using the Longley and Hald datasets. For the Longley dataset, Cook’s distance was applied to estimated coefficients, fitted values, residuals, and leverage points. For the Hald dataset, Cook’s distance was used to assess estimated coefficients, fitted values, and mean squared error (MSE), while bias was also examined. The results demonstrated the effectiveness of the proposed estimators. A Monte Carlo simulation study evaluates the results of TLRPLSEs using Cook’s distance across various influential cases at different multicollinearity levels and sample sizes. The results indicate that TLRPLSEs outperform competing estimators, maintaining robustness against multicollinearity and influential cases.