| Generalized linear model was proposed by Nelder and Wedderbum in 1972.It is a natural generalization of linear models,and its application value is more extensive.This paper mainly studies the following generalized linear models yi=h(xiTβ)+ei,i=1,2,…,n,where yi is response vector,xi is the explanatory variable,h(·)is a continuous differentiable function,β is an unknown parameter vector of d dimension,ei is the error term.Although there are a lot of results of the GLM,no one has used penalized Lq-likelihood method to study variable selection and parameter estimation of the GLM(as far as I know).Therefore,based on the penalized Lq-likelihood method and generalized linear regression,the following problems are studied in the paper.In Chapter 2,we consider the GLM with independent errors.We propose a penalized Lq-likelihood estimation method(PLqE),which can select variables and estimate parameters at the same time,and prove that the PLqE method has Oracle properties.Moreover,we give some related numerical simulation examples,which shows that PLqE is robust,while the penalized likelihood estimation is not robust.In Chapter 3,the PLqE method is applied to the generalized linear regression model with first order autoregressive errors(AR(1)),variable selection and parameter estimation are carried out at the same time.We discuss its Oracle properties and give the corresponding simulation applications,it is shown that our method is not only robust,but also superior to PLE in the sense of the mean square errors.In Chapter 4,we theoretically prove the robustness of the PLqE by the influence function,and then verify its robustness by simulation examples.At the same time,we preliminarily study the relationship between the influence function and q from the aspects of theory and simulation examples,so as to select the appropriate q value. |
