‎One of the factors affecting the statistical analysis of the data is the presence of outliers‎. ‎The methods which are not affected by the outliers are called robust methods‎. ‎Robust regression methods are robust estimation methods of regression model parameters in the presence of outliers‎. ‎Besides outliers‎, ‎the linear dependency of regressor variables‎, ‎which is called multicollinearity‎, ‎the large number of regressor variables with respect to sample size‎, ‎specially in high dimensional sparse models‎, ‎are problems which result in efficiency reduction of inferences in classical regression methods‎. ‎In this paper‎, ‎we first study the disadvantages of classical least squares regression method‎, ‎when facing with outliers‎, ‎multicollinearity and sparse models‎. ‎Then‎, ‎we introduce and study robust and penalized regression methods‎, ‎as a solution to overcome these problems‎. ‎Furthermore‎, ‎considering outliers and multicollinearity or sparse models‎, ‎simultaneously‎, ‎we study penalized-robust regression methods‎. ‎We examine the performance of different estimators introdused in this paper‎, ‎through three different simulation studies‎. ‎A real data set is also analyzed using the proposed methods‎.

‎The most important goal of statistical science is to analyze the real data of the world around us‎. ‎If this information is analyzed accurately and correctly‎, ‎the results will help us in many important decisions‎. ‎Among the real data around us which its analysis is very important‎, ‎is the water consumption data‎. ‎Considering that Iran is located in a semi-arid climate area of the earth‎, ‎it is necessary to take big steps for predicting and selecting the best and the most appropriate accurate models of water consumption‎, ‎which is necessary for the macro-national decisions‎. ‎But analyzing the real data is usually complicated‎. ‎In the analysis of the real data set‎, ‎we usually encounter with the problems of multicollinearity and outliers points‎. ‎Robust methods are used for analyzing the datasets with outliers and ridge method is used for analyzing the data sets with multicollinearity‎. ‎Also‎, ‎the restriction on the models is resulted from using non-sample information in estimation of regression coefficients‎. ‎In this paper‎, ‎it is proceeded to model the water consumption data using robust stochastic restricted ridge approach and then‎, ‎the performance of the proposed method is examined through a Monte Carlo simulation study‎.