SICA Penalized Estimator Possessing The Oracle Property With A Diverging Number Of Parameters

An important issue in Statistical modeling is how to select sig-nificant variables from a large number of explanatory variables. This is a topic about variable selection. Up till now, there have been a lot of researches on variable selection, and among them, the penalty function method is a fairly popular as well as of great applicability approach. This method can simultaneously accomplish variable selec-tion and regression coefficients estimation by minimizing an objective function that is composed of a loss function plus a penalty function.Lv and Fan (2009) propose a variable selection method based on the SICA penalty function. Using penalized least squares, they show the weak oracle property of their estimators with the dimensionality growing exponentially with sample size. we establish the consistency and oracle property of SICA penalized least squares estimators with a diverging number of parameters.Furthermore, we generalize the above research work into the variable selection problem for Cox's proportional hazards model in survival analysis. Using penalized partial likelihood method, i.e.. the negative logarithm of partial likelihood function added to SICA penalty function, we establish the consistency and oracle property of SICA penalized partial likelihood estimators with a diverging number of parameters. About algorithm issues, we first smooth the objec-tive function, and then solve the problem by using the quasi-Newton method combined with line search technique. Quasi-Newton method only require the first derivative of the objective function, so it can avoid a large number of calculations; the line search technique can guarantee the convergence of a wide range, so it can avoid the choice of initial values influence convergence of the algorithm. Our algo-rithm strategy converges quickly, with superlinear convergence rate. The simulation studies showed that our proposed method can improve the accuracy and efficiency of variable selection comparing with the usual penalized methods, such as LASSO. In addition, we use the pro-posed method to analyze the PBC (primary biliary cirrhosis) data, and the results of the analysis are consistent with the existing con-clusions.