Study On The Parameter Estimation And Robust Variable Selection For Linear Model

Parameter estimation is one of the most important parts of building statistic model,which is closely related to the accuracy of models and leads many researchers to devote themself into improving it.There are many practical ways to estimate parameter for norm data,while our job is to focus on how to estimate parameter correctly in this situation where there exists multicollinearity and outliers simultaneously in the data or the data is coming from heavy-tailed distribution.We develop our researching work for parameter estimation in linear model from the following two data conditions.In the first place,we consider that all the explanatory variables are important but there exists multicollinearity and outliers simultaneously in the data.After learning from the previous estimations,we propose a new estimation to cope with this data situation.The new method has the nice qualities of ridge estimation,two-parameter estimation,and robust estimation simultaneously then it can overcome the negative effects from multicollinearity and outliers in the data.We theoretically compare our new method RTRME with the previous two estimations RTRE and TRME under the Mean Square Error criterion.We have done several numerical simulations under different parameter situations,which shows that our new estimation almost outperforms the other two estimators RTRE and TRME for most of time.And the following real data example also proves that our new method is better than the other two estimations.All the outcomes convince that our new estimation RTRME can be applied in practice.In another case,we suppose that there exists redundant regressor variables along with multicollinearity and outliers simultaneously in the data.In this situation,we resort to robust variable selection to estimate parameter and identify proper model for such data.Firstly,we introduced the traditional robust variable selection method— LAD-lasso and robust estimator---PWLS.In order to take advantage of the two method's good properties,we propose a new robust variable selection,the PWLS-LAD.Then we mainly discuss the deriving thoughts and the practical algorithm about the new method,moreover,we have discussed how to choose the tunning parameters.As the first case,we also have done numerical simulations to illustrate our new method's merits by comparing with conventional variable selection methods of LASSO,SCAD,LAD-lasso under three cases,the norm data without outliers,contaminated data with outliers,heavy-tailed data,respectively.We mainly concentrate on the following aspects including the accuracy of picking the right non-zero parameter variables,the predictive errors of the models and the MSE values of all the parameter estimations.From the numerical results,we find that there is no remarkable difference between our new method and the traditional ways in norm data case.However,the robust methods---LAD-lasso and PWLS-LAD,are better than the two non-robust ways when outliers appearing in the data,more importantly,our new method PWLS-LAD outperforms the LAD-lasso for most of cases even in data with outliers or heavy-tailed data.Therefore,we can say that our new method is meaningful and robust which can be applied in wide data situations both in norm data and extreme data.