Penalized Lq-likelihood estimator and its influence function in generalized linear models

Consider the following generalized linear model (GLM) \(\begin{aligned} y_i=h(x_i^T\beta )+e_i,\quad i=1,2,\ldots ,n, \end{aligned}\) $$\begin{array}{c}\hfill y_i=h(x_i^T\beta )+e_i,i=1,2,\dots ,n,\end{array}$$ where h(.) is a continuous differentiable function, \(\{e_i\}\) $\{e_i\}$ are independent identically distributed (i.i.d.) random variables with zero mean and known variance \(\sigma ^2\) ${\sigma }^2$ . Based on the penalized Lq-likelihood method of linear regression models, we apply the method to the GLM, and also investigate Oracle properties of the penalized Lq-likelihood estimator (PLqE). In order to show the robustness of the PLqE, we discuss influence function of the PLqE. Simulation results support the validity of our approach. Furthermore, it is shown that the PLqE is robust, while the penalized maximum likelihood estimator is not.