Smoothed Quantile Regression for Spatial Data

Music City Center 

Existing methods for spatial data either have difficulty capturing heterogeneous patterns over complex domains or overlook the heterogeneity in the tail of the response distribution. We introduce a flexible quantile spatial model framework, which can simultaneously capture spatial nonstationarity and heterogeneity via constant and spatially varying coefficients. It also allows researchers to study patterns across different tails of the response distribution. We propose a smoothed quantile bivariate triangulation method based on penalized splines on triangulation and convolution smoothing in the quantile loss. The developed method can effectively capture spatial nonstationarity while preserving critical data features such as shape and smoothness across complex and irregular domains. Under some regularity conditions, we show that the proposed estimator can achieve an optimal convergence rate under the L2-norm. In addition, we establish the Bahadur representation of the estimator, which allows us to establish the asymptotic normality for the constant coefficient estimator and construct asymptotic confidence intervals. To improve finite-sample performance, we also consider a wild bootstrap method for constructing confidence intervals. Through simulation studies, we demonstrate the numerical and computational advantages of the proposed estimator over existing methods. The application of the proposed method to study the spatial heterogeneity of US mortality demonstrates that the mortality rates depend on socioeconomic factors differently across space and the tails of the mortality distribution. This is joint work with PhD student Jilei Lin and his advisor, Dr. Judy Wang, at The George Washington University, as well as Dr. Myungjin Kim from Kyungpook National University.