A fresh view on Least Quantile of Squares Regression based on new optimization approaches

Regression analysis is an important instrument to determine the effect of the explanatory variables on response variables. When outliers and bias errors are present, the standard weighted least squares estimator may perform poorly. For this reason, many alternative robust techniques have been studied in literature. In these terms, the Least Squares Quantile (LQS), and in particular the Least Squares Median, are among the regression estimators that exhibit better robustness properties. However, the accurate computation of this estimators is computationally demanding, resulting in a difficult estimator to obtain. In this paper, new novel approaches to compute a global optimal solution for the LQS estimator based on single-level and bilevel optimization methods are proposed. An extensive computational study is provided to support the efficiency of the methods considered, and an ad hoc procedure to address the scalability of the problem to larger instances is proposed.