Variable Selection For Single-index Model With M-estimated

Robust parameter estimation and variable selection are the two research priorities in the statistics. In the regression analysis, these two issues need to be studied in the specific model. From linear regression to non-parametric regression then to semi-parametric regression, various model emerging. The single index model is an important semi-parametric regression model, not only for its powerful ability to describe data, but also has profound theoretical significance.In order to select the variable during the same time to overcome effects of outlier and data errors with a heavy tail distribution, research was focused on the quantile regression(QR) in recent years. But the QR has two disadvantages. First, the confidence field of estimated parameters is larger 25%~30% than the least squares method; Secondly, due to the loss function of quantile regression is not differentiable on origin, the calculation is more complicated. In this paper, we study the variable selection problem based on M-estimated.In the introduction, we describes the single index model and the partial linear model single index, as well as technology and research status. In the second chapter, a method of variable selection for single-index model is proposed, which is based on the M-estimation and the adaptive LASSO. And its oracle property is established and proved. Unlike the existing M-estimator of the single-index model, the unknown link function is approximated by B-spline. Simulations with various non-normal errors and a real data analysis are conducted to assess the finite sample property of the proposed estimation and variable selection methods. In the third chapter, a method of variable selection for partial single-index model is proposed, which is based on the M-estimation and the adaptive LASSO. And its oracle property is established and proved. Unlike the existing M-estimator of the single-index model, the unknown link function is approximated by B-spline. Simulations with various non-normal errors and a real data analysis are conducted to assess the finite sample property of the proposed estimation and variable selection methods.