Robust Liu Estimator Used to Combat Some Challenges in Partially Linear Regression Model by Improving LTS Algorithm Using Semidefinite Programming

Outliers are a common problem in applied statistics, together with multicollinearity. In this paper, robust Liu estimators are introduced into a partially linear model to combat the presence of multicollinearity and outlier challenges when the error terms are not independent and some linear constraints are assumed to hold in the parameter space. The Liu estimator is used to address the multicollinearity, while robust methods are used to handle the outlier problem. In the literature on the Liu methodology, obtaining the best value for the biased parameter plays an important role in model prediction and is still an unsolved problem. In this regard, some robust estimators of the biased parameter are proposed based on the least trimmed squares (LTS) technique and its extensions using a semidefinite programming approach. Based on a set of observations with a sample size of n, and the integer trimming parameter h <= n, the LTS estimator computes the hyperplane that minimizes the sum of the lowest h squared residuals. Even though the LTS estimator is statistically more effective than the widely used least median squares (LMS) estimate, it is less complicated computationally than LMS. It is shown that the proposed robust extended Liu estimators perform better than classical estimators. As part of our proposal, using Monte Carlo simulation schemes and a real data example, the performance of robust Liu estimators is compared with that of classical ones in restricted partially linear models.