New technique for high dimensional data : robust linear regression using L1-penalized mm-estimation

Large datasets, where the number of predictors p is larger than the sample sizes n, have become very popular in recent years. These datasets pose great challenges for building a linear good prediction model. In addition, when dataset contains a fraction of outliers and other contaminations, linear regression becomes a difficult problem. Therefore, we need methods that are sparse and robust at the same time. In this thesis, we employed the approach of MM estimation and proposed L1-Penalized MM-estimation (MM-Lasso) as a new estimation method. Our proposed estimator uses sparse LTS estimator as initial estimator to compute penalized M-estimator getting sparse modeli estimation with high breakdown value and good prediction. We implemented MM-Lasso by using C programming language and calling it from R package. To evaluate our proposed estimator, we extended the SimFrame R package, which is a general framework for simulation studies in statistics. We generated three data models to represent low, moderate and high dimensional data. We also implemented the function for generating the data for the contamination. Simulation study shows that the MM-lasso estimation has better prediction performance than its competitors in the presence of leverage points.