Quadratic mixed integer programming and support vectors for deleting outliers in robust regression

We consider the problem of deleting bad influential observations (outliers) in linear regression models. The problem is formulated as a Quadratic Mixed Integer Programming (QMIP) problem, where penalty costs for discarding outliers are used into the objective function. The optimum solution defines a robust regression estimator called penalized trimmed squares (PTS). Due to the high computational complexity of the resulting QMIP problem, the proposed robust procedure is computationally suitable for small sample data. The computational performance and the effectiveness of the new procedure are improved significantly by using the idea of e-Insensitive loss function from support vectors machine regression. Small errors are ignored, and the mathematical formula gains the sparseness property. The good performance of the e-Insensitive PTS (IPTS) estimator allows identification of multiple outliers avoiding masking or swamping effects. The computational effectiveness and successful outlier detection of the proposed method is demonstrated via simulated experiments.