Estimation and Inference of Quantile Spatially Varying Coefficient Models Over Complicated Domains

This paper presents a flexible quantile spatially varying coefficient model (QSVCM) for the regression analysis of spatial data. The proposed model enables researchers to assess the dependence of conditional quantiles of the response variable on covariates while accounting for spatial nonstationarity. Our approach facilitates learning and interpreting heterogeneity in spatial data distributed over complex or irregular domains. We introduce a quantile regression method that utilizes bivariate penalized splines in triangulation to estimate unknown functional coefficients. We establish the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msub> </mml:math> convergence of the proposed estimators, demonstrating their optimal convergence rate under certain regularity conditions. An efficient optimization algorithm is developed using the alternating direction method of multipliers (ADMM). We develop wild residual bootstrap-based pointwise confidence intervals for the QSVCM quantile coefficients. Furthermore, we construct reliable conformal prediction intervals for the response variable using the proposed QSVCM. Simulation studies show the remarkable performance of the proposed methods. Lastly, we illustrate the practical applicability of our methods by analyzing the mortality dataset and the supplementary particulate matter (PM) dataset in the United States.