Title: Predictive risk estimation in high-dimensional misspecified quantile regression
Authors: Alexander Giessing - Princeton University (United States) [presenting]
Abstract: Predictive modeling plays an important role in many scientific disciplines, including finance and economics. A natural way to gauge predictive models is to estimate their predictive risk. We develop a framework for estimating the predictive risk of high-dimensional quantile regression (QR) models under arbitrary misspecifications. In this framework, the marginal distributions of the response and predictor variables can be non-Gaussian and their relationship can be linear, nonlinear or nonparametric. Our estimator of the predictive risk is based a novel characterization of the down-ward bias of the in-sample risk of the QR fit. In particular, we show that the down-ward bias depends on the specification of the QR model, the covariance of the predictor variables and the conditional density of the response variable. Based on this characterization we propose a de-biased nonparameteric plug-in estimator for the predictive risk and establish its uniform consistency and asymptotic normality. On the theoretical side we provide a new strong Bahadur representation for misspecified QR models with a diverging number of unbounded predictor variables. This new representation fully accounts for any sort of misspecification and therefore retains fast rates of convergence of the remainder term even if the misspecification persists as the sample size increases.