Several Penalty Likelihood Estimation In The Nature Of The Generalized Linear Model

Model selection is fundamental to the statistical analysis of high-dimensional models. The traditional approaches of stepwise regression and subset selection method not only ignore stochastic errors in the selection of variables, but also suffer tremendous computing workload in high-dimensional models. The method of LASSO was proposed to overcome these limitations. Then the thought of penalized likelihood, which simultaneously selects variables and estimates relevant coefficients, was presented based on LASSO. Many approaches have been proposed and studied in the framework of penalized likelihood, such as SCAD、adaptive LASSO、 elastic net and relaxed LASSO.The main jobs of this paper are following:(1) We summarize the development and current situation of penalized likelihood. Then we systematically introduce the idea of penalized likelihood in view of linear model and state the penalty functions of SCAD、 adaptive LASSO、 elastic net and relaxed LASSO along with two popular algorithms;(2) We provide a general form of penalized likelihood estimation in view of generalized linear model. And we study the asymptotic properties of the adaptive LASSO and SCAD in the logistic regression model. Furthermore we show that under some certain conditions, the adaptive LASSO and SCAD estimators of logistic regression models enjoy the oracle properties.