Neural Networks for Partially Linear Quantile Regression

Abstract

Deep learning has enjoyed tremendous success in a variety of applications but its application to quantile regressions remains scarce. A major advantage of the deep learning approach is its flexibility to model complex data in a more parsimonious way than nonparametric smoothing methods. However, while deep learning brought breakthroughs in prediction, it often lacks interpretability due to the black-box nature of multilayer structure with millions of parameters, hence it is not well suited for statistical inference. In this paper, we leverage the advantages of deep learning to apply it to quantile regression where the goal to produce interpretable results and perform statistical inference. We achieve this by adopting a semiparametric approach based on the partially linear quantile regression model, where covariates of primary interest for statistical inference are modelled linearly and all other covariates are modelled nonparametrically by means of a deep neural network. In addition to the new methodology, we provide theoretical justification for the proposed model by establishing the root-$n$ consistency and asymptotically normality of the parametric coefficient estimator and the minimax optimal convergence rate of the neural nonparametric function estimator. Across several simulated and real data examples, our proposed model empirically produces superior estimates and more accurate predictions than various alternative approaches.