Asymptotic Theory Of Variable Selection Under Nonconvex Penalty In Generalized Linear Models With Adaptive Designs

Variable selection is a basic preprocessing method for high-dimensional statisti-cal models,which has an important meaning in statistics.In the process of modeling,if the independent variables are included,it will increase the difficulty of computing and affect the accuracy of parameter estimation.While we miss a significant variable or a model is misset,it will seriously affect the results of the model analysis,making the model lose its interpretability,so variable selection becomes very important.The generalized linear models(GLMs),which was first introduced by Nelder and Wedderburn(1972),is an important extension of the general linear model,and can be applied to both continuous and discrete cases.Wedderburn(1974)proposed if the dis-tribution of the response variables is unknown,while the mean and variance structure are known,the quasi likelihood equation can be constructed by imitating the maxi-mum likelihood method in GLMs.In this paper,the variable selection of GLMs with adaptive designs are discussed,we use the quasi likelihood method combined with the non-convex penalty.This method can estimate parameters and select variables simultaneously.We choose the SCAD penalty function proposed in Fan and Li(2001),The quasi likelihood equation with SCAD penalty function in fixed designs and adaptive designs are introduced,we also prove the oracle properties of the quasi likelihood estimator under adaptive de-signs.In simulation,we calculate the ratio of the real model and the real 0 coefficient variables were selected.Through the simulation results,we can see that this method can select these variables which coefficients are 0 under adaptive designs.With the increase of sample size,the true model can be selected.