Bivariate spatial process modeling for constructing indicator or intensity weighted spatial CDFs

A spatial cumulative distribution function (SCDF) gives the proportion of a spatial domain D having the value of some response variable less than a particular level w. This article provides a fully hierarchical approach to SCDF modeling, using a Bayesian framework implemented via Markov chain Monte Carlo (MCMC) methods. The approach generalizes the customary SCDF to accommodate density or indicator weighting. Bivariate spatial processes emerge as a natural approach for framing such a generalization. Indicator weighting leads to conditional SCDFs, useful in studying, for example, adjusted exposure to one pollutant given a specified level of exposure to another, Intensity weighted (or population density weighted) SCDFs are particularly natural in assessments of environmental justice, where it is important to determine if a particular sociodemographic group is being excessively exposed to harmful levels of certain pollutants. MCMC methods (combined with a convenient Kronecker structure) enable straightforward estimates or approximate estimates of bivariate, conditional, and weighted SCDFs. We illustrate our methods with two air pollution datasets, one recording both nitric oxide (NO) and nitrogen dioxide (NO₂) ambient levels at 67 monitoring sites in central and southern California, and the other concerning ozone exposure and race in Atlanta, GA.