Bridge Estimator For Cox’s Proportional Hazard Model

We investigate the variable selection problem for Cox’s proportional hazards model, andpropose a unified model selection and estimation procedure with desired theoretical propertiesand computational convenience.The new method is based on a penalized log partial likelihood with the bridge penalty onregression coefficients. We propose bridge estimator in Cox’s proportional hazard modelfor parameter estimation and variable selection(i.e., to distinguish between covariates whosecoefficients are exactly zero and covariates whose coefficients are nonzero). Under reasonableconditions the consistency of the bridge estimator can be achieved. Furthermore, it can selectthe nonzero coefficients with a probability converging to1and the estimators of nonzerocoefficients have the asymptotic normality, namely the oracle property.This paper has the following several chapters:Chapter1is mainly introduces the form of Cox proportional hazards method and thedefinition of penalty functions, the form of partial likelihood method and mang variableselected methods. Corresponding to the kinds of penalized functions, we propose thescholars at home and abroad on the basis of research achievements.In the second of chapter, we give the definition of penalized partial likelihood function viabridge estimator isUnder reasonable conditions, we show that penalized empirical likelihood can select thenonzero coefficients with a probability converging to1and the estimators of nonzerocoefficients have the asymptotic normality.All the proofs are given in the chapter3. Lagrange multiplier method, central limit theoremand the method on estimation of the order of random variables is applied to the proof of thetheorems in the process.In chapter4outlines computational aspect of bridge estimator via LQA and LLAalgorithms. We have examined the LQA and LLA algorithms and those two methods havesimilar results. In chapter5, we present numerical simulations. All the numerical simulations researchverify the correctness of theorems. Comparison of EL, LASSO, SCAD, ALASSO andbridge.