  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.
