# Smooth Density Spatial Quantile Regression

## TL;DR

This paper introduces a flexible spatial quantile regression model combining spline and Pareto tail distributions, enabling detailed analysis of covariate effects on distributional extremes and non-stationary covariance functions.

## Contribution

It develops a novel model-based approach for spatial quantile regression that handles heavy tails and non-stationarity, with proven differentiability and efficiency improvements.

## Key findings

- More efficient estimates in heavy-tailed distributions
- Effective modeling of non-stationary covariance functions
- Application to real-world benzene measurement data

## Abstract

We derive the properties and demonstrate the desirability of a model-based method for estimating the spatially-varying effects of covariates on the quantile function. By modeling the quantile function as a combination of I-spline basis functions and Pareto tail distributions, we allow for flexible parametric modeling of the extremes while preserving non-parametric flexibility in the center of the distribution. We further establish that the model guarantees the desired degree of differentiability in the density function and enables the estimation of non-stationary covariance functions dependent on the predictors. We demonstrate through a simulation study that the proposed method produces more efficient estimates of the effects of predictors than other methods, particularly in distributions with heavy tails. To illustrate the utility of the model we apply it to measurements of benzene collected around an oil refinery to determine the effect of an emission source within the refinery on the distribution of the fence line measurements.