The Nature Of The Estimated Amount Of Penalty Under The Generalized Linear Model

Variable selection is fundamental to high-dimensional statistical modeling. Traditional approaches in use are stepwise selection procedures, which can be computationally expensive and ignore stochastic errors in the variable selection process. Hence againsting for the limitations of the traditional methods, pe-nalized likelihood approaches are proposed to handle these kind of problems. The proposed methods select variables and estimate coefficients simultaneously. They are readily applied to a variety of parametric models. They can also be applied easily to nonparametric modeling by using wavelets and splines.This dissertation studies the penalized likelihood approaches and the prop-erties of some penalty functions. The main jobs are following:(1) We summarize the development of the penalized likelihood approaches and systematically introduce the unified framework. And we give the form of some penalty functions on linear model.(2) In the researches of the penalized likelihood methods in linear model on the basis of our predecessors, the penalized likelihood approaches are introduced into the generalized linear model. And we study the asymptotic properties of the Adaptive Lasso and SCAD in Poisson log-linear regression model. Under some mild conditions, we show that Adaptive Lasso and SCAD estimators are sparse and asymptotically normal. In the other words, they enjoy oracle property.(3) We finally point out some problems to further study.