Quantile regression for varying-coefficient partially nonlinear models with randomly truncated data

This paper is concerned with quantile regression (QR) inference of varying-coefficient partially nonlinear models where the response is subject to randomly left truncation. A three-stage estimation procedure for parameter and coefficient functions is proposed based on the weights which are random quantities and determined by the product-limit estimates of the distribution function of truncated variable. The asymptotic properties of the proposed estimators are established. Further, a variable selection procedure is developed by combining the quantile loss function with the adaptive LASSO penalty to get sparse estimation of the parameter. The proposed penalized QR estimators are shown to possess the oracle property. In addition, a bootstrap-based test procedure is proposed via an extended generalized likelihood ratio test statistic to check whether the coefficient function has a specific parametric form. Both simulations and real data analysis are conducted to demonstrate the proposed methods.