The penalized least square method is a popular method to deal with high dimensional data ,where the number of explanatory variables is large than the sample size . The properties of penalized least square method are given high prediction accuracy and making estimation and variables selection
At once. The penalized least square method gives a sparse model ,that meaning a model with small variables so that can be interpreted easily .The penalized least square is not robust ,that means very sensitive to the presence of outlying observation , to deal with this problem, we can used a robust loss function to get the robust penalized least square method ,and get robust penalized estimator and it can deal problems of dimensions and outliers .In this paper a compression had been made Sparse LTS estimator and MM Lasso by using simulation and the simulation results show that the MM Lasso is best for every experiments, Depending on the criteria for the Mean Square Error, False Positive Rate and False negative Rate .
