Variable Selection for Additive Partial Linear Quantile Regression with Missing Covariates

TL;DR

This paper develops a variable selection method for additive partial linear quantile regression models with missing covariates, using inverse probability weighting and non-convex penalties, with theoretical guarantees and practical validation.

Contribution

It introduces a novel approach combining weighted estimation, non-convex penalties, and handling missing data in quantile regression with partial linear models, including high-dimensional settings.

Findings

The proposed estimator achieves the oracle property.

Simulation studies demonstrate superior variable selection accuracy.

Application to real data illustrates practical effectiveness.

Abstract

The standard quantile regression model assumes a linear relationship at the quantile of interest and that all variables are observed. We relax these assumptions by considering a partial linear model while allowing for missing linear covariates. To handle the potential bias caused by missing data we propose a weighted objective function using inverse probability weighting. Our work examines estimators using parametric and nonparametric estimates of the missing probability. For variable selection of the linear terms in the presence of missing data we consider a penalized and weighted objective function using the non-convex penalties MCP or SCAD. Under standard conditions we demonstrate that 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 non-convex penalty function. The performance of the method is evaluated using Monte Carlo simulations and our methods are applied to model amount of time sober for patients leaving a rehabilitation center.