Estimation And Variable Selection Of Single Index Quantile Regression Model With Response Missing

In the fields of surveys and experimental research,data are often missed due to investigators’ information input errors or date unobservability.In many applications,the data often suffer from significant heteroscedasticity or the distribution of sharp peaks,thick tails.At the same time,the true model is often high sparse,and so we need to remove the covariates unrelated to the response variable.Therefore,in this article we study estimation and variable selection problems for the single-index quantile regression model with response variable missed.For the estimation of the single-index quantile regression model with missing response variables,we first use the B-spline technique to approximate the unknown index function,and then construct the inverse probability weighted quantile regression loss function;Finally,the parameters and function estimates are obtained through the nonlinear optimization method.The effectiveness of the proposed method is demonstrated through the numerical simulation and the analysis of the Boston housing price example.In the applications of quantile regression models,we tend to collect as many variables as possible to explain the response variable,so as to establish a high-dimensional quantile regression model.However some of the covariates unsignificantly effect the response variable.Therefore,we need to select important variables for the highdimensional quantile regression model.In the variable selection,we choose adaptive LASSO to establish an penalized inverse probability weighted loss function.By minimizing such loss function,we can select important variables.We use the BIC criterion to select the penalty parameters.Numerical studies show that our proposed variable selection method perform very well for moderate sample.