By evolving science, knowledge, and technology, new and precise methods for measuring, collecting, and recording information have been innovated, which have resulted in the appearance and development of high-dimensional data. The high-dimensional data set, i.e., a data set in which the number of explanatory variables is much larger than the number of observations, cannot be easily analyzed by traditional and classical methods, the same as the ordinary least-squares method, and its interpretability will be very complex. Although in classical regression analysis, the ordinary least-squares estimation is the best estimation method if the essential assumptions are met, it is not applicable for high-dimensional data, and in this condition, we need to apply the modern methods. In this research, it is first mentioned the drawbacks of classical methods in the analysis of high-dimensional data and then, it is proceeded to introduce and explain the modern and common approaches of the regression analysis for high-dimensional data same as principal component analysis and penalized methods. Finally, a simulation study and real-world data analysis are performed to apply and compare the mentioned methods in high-dimensional data.

‎The method of least squares is a very simple‎, ‎practical and useful approach for estimating regression coefficients of the linear models‎. ‎This statistical method is used by users of different fields to provide the best unbiased linear estimator with the least variance‎. ‎Unfortunately‎, ‎this method will not have reliable output if outliers are present in the dataset‎, ‎as the collapse point (estimator consistency criterion) of this method is 0% ‎. ‎It is therefore important to identify these observations‎. Until now, ‎the various methods have been proposed to identify these observations‎. ‎In this article‎, the proposed methods are ‎reviewed ‎and ‎discussed in details‎‎‎. ‎Finally‎, ‎by presenting a simulation example‎, ‎we examine each of 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.

Analysis and modeling the‎ ​‎h​igh-dimensional data is one of the most challenging problems faced by the world nowaday‎. ‎Interpretation of such data is not easy and needs to be applied to modern methods‎. The penalized methods are one of the most popular ways to analyze the high-dimensional data‎. ‎Also‎, ‎the regression models and their analysis are affected by the outliers seriously‎. The least trimmed squares method is one of the best robust approaches to solve the corruptive influence of the outliers‎. ‎Semiparametric models‎, ‎which are a combination of both parametric and nonparametric models‎, ‎are very flexible models‎. ‎They are useful when the model contains both parametric and nonparametric parts‎. ‎The main purpose of this paper is to analyze semiparametric models in high-dimensional data with the presence of outliers using the robust sparse Lasso approach‎. ‎Finally‎, ‎the performance of the proposed estimator is examined using a real data analysis about production of vitamin B2‎.