Quadratic Mixed Integer Programming Models in Minimax Robust Regression Estimators

The robust estimation of regression parameters is formulated in terms of mixed integer-quadratic programming problem. The main contribution of this technique is that it improves the estimator efficiency by down-weighting only bad influential points, either y-outliers or x-outliers. We follow the minimax strategy where the objective function of our mathematical programming formulation is mainly a Huber loss function, and bad influential outliers pulled towards the regression line with low cost. This penalized pulling cost is a function of Mallows type weights, and in the modified data a GM estimator (Schweppe type) could be defined. The main advantage of the proposed technique is that data points are not down-weighted, unless they have increased substantially the square residuals. Previously published mixed integer formulations withdraw data points, the most influential even if they are not bad influential points. GM estimators are compared to our proposal via simulated experiments, the robust estimator obtained by quadratic programming is reasonable.