We consider the problem of identifying multiple outliers in linear regression models. In robust regression the unusual observations should be removed from the sample in order to obtain better fitting for the rest of the observations. Based on the LTS estimate, we propose a penalized trimmed square estimator PTS, where penalty costs for discarding outliers are inserted into the loss function. We search for suitable penalty costs for multiple high-leverage outliers, which are based on robust leverage and scale. Thus, the best fit for the majority of the data is obtained after eliminating only outliers from the data set. The robust estimation is obtained by minimizing the loss function with a mathematical programming technique, computationally suitable for small sample data. The computational load and the effectiveness of the new procedure are improved by using the idea of e-insensitive tube from support vectors machine regression. The PTS loss function is transformed to an e-Insensitive, where small errors are ignored, and the mathematical formula gains the sparseness property. The good performance of the PTS estimator allows identification of multiple outliers avoiding masking or swamping effects. We conduct benchmark examples and a simulation study to investigate the procedure’s efficiency for robust estimation and power as outlier detection. As a result, the performance of both types of PTS is superior to other methods and is worth the extra computational load.

## Citation

Department of Mathematical and Physical Sciences School of Engineering Aristotle University of Thessaloniki Greece

## Article

View A Penalized Trimmed Squares Method for Deleting Outliers in Robust Regression