Estimators And Variable Selection Methods For Linear Models Under Multicollinearity And Heteroscedasticity

Linear regression model is one of the earliest development and most widely used model in modern mathematical statistics, which plays a central part in modern statistical methods. The parameter estimation and variable selection are both very important areas of research among the linear regression models. Many scholars have made great contributions in these two areas, have found a lot of useful methods. As to parameter estimation, the most classic is the ordinary least squares estimate, but it perfects well only under many assumptions. While with more and more widely application of the statistics, the practical models involved in applications are more and more complicated.We should not only solve the problems of variable selection and parameter estimation in high dimensional linear models, but also should make efforts on other generalized linear models. For different estimates, different criteria can be used to decide which one is better for different needs of study.In the presence of multicollinearity of the design matrix, series of biased estimation have been proposed. This paper studies the almost unbiased Liu estimates, consider the admissibility of the almost unbiased Liu estimator to the ordinary least squares estimator in the linear regression model in the Fisherian and Mahalanobis loss functions.The results show that the almost unbiased Liu estimator perfects better than the ordinary least squares under some assumptions, while is inadmissible under the Mahalanobis loss function. Also, a simulation study is given to show the theoretical results.At the same time, when the errors of linear model are heteroscedasticity, according to the existing researches of variable selection and parameter estimation, the weighted adaptive elastic net is put forward to variable selection and parameter estimation, it shows that when the variance is known the estimator owns the oracle property by theoretical derivation, at the end of the chapter some numerical simulations are also made to illustrate the theoretical results.