Smoothed Quantile Regression for Spatial Data

Existing methods for spatial data often struggle to capture heterogeneous patterns over complex domains or ignore heterogeneity in the tails of the response distribution. We introduce a quantile spatial model framework that accommodates both spatial nonstationarity and tail heterogeneity through constant and spatially varying coefficients. We propose a smoothed quantile bivariate triangulation (SQBiT) method based on penalized splines on triangulation and convolution smoothing of 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. Simulations highlight the numerical and computational advantages of SQBiT over existing methods. Applying SQBiT to U.S. mortality data reveals how socioeconomic factors influence mortality rates differently across spatial regions and distribution tails. An R package implementing SQBiT is available on GitHub.