Variable Selection and Its Asymptotic Theory Based on L_\gamma Penalty in Generalized Linear Models with Adaptive Designs

In this paper, L_\gamma penalty method proposed by Frank and Friedman\ucite3 is used to study the variable selection problem and its asymptotic properties based on the penalized quasi-likelihood method in generalized linear models with adaptive designs. This method can perform parameter estimation and variable selection simultaneously. For 0<\gamma<1, the existence, consistency and Oracle properties of the estimators based on L_\gamma penalty and the quasi-likelihood method in generalized linear models with adaptive designs are proved under appropriate conditions. These results generalize the related theories of generalized linear models from the case of fixed designs to the case of adaptive designs. The validity of our obtained theory is verified by numerical simulation and real data analysis inthis paper.