Variable selection for additive partial linear quantile regression with missing covariates

The standard quantile regression model assumes a linear relationship at the quantile of interest and that all variables are observed. These assumptions are relaxed by considering a partial linear model with missing covariates. A weighted objective function using inverse probability weighting is proposed to remove the potential bias caused by missing data. Estimators using parametric and nonparametric estimates of the probability an observation has fully observed covariates are examined. A penalized and weighted objective function using the nonconvex penalties MCP or SCAD is used for variable selection of the linear terms in the presence of missing data. Assuming the missing data problems remains a low dimensional problem the penalized estimator has the oracle property including cases where p≫n. Theoretical challenges include handling missing data and partial linear models while working with a nonsmooth loss function and a nonconvex penalty function. The performance of the method is evaluated using Monte Carlo simulations and the methods are applied to model amount of time sober for patients leaving a rehabilitation center.