Nonparametric construction of probability maps under local stationarity

Abstract

The environmental contamination risk can be evaluated in a specific area by approximating the probability that the pollutant under study exceeds a critical value. This issue requires the estimation of the distribution function involved, which can be addressed by applying the indicator kriging methodology or by approximating the sill of the variogram of the underlying indicator process. These approaches demand an appropriate characterization of the indicator variogram, which in turn requires a previous specification of the trend function, if the latter is suspected to be non-constant. Since accuracy of the results will be strongly dependent on the adequate approximation of both functions, we suggest proceeding in a different way to avoid these requirements. Thus, in the current paper, two kerneltype estimators are proposed, based on first approximating the distribution at the sampled sites and then obtaining a weighted average of the resulting values, to derive a valid estimator at each (sampled or unsampled) location. Consistency of the kernel approaches is proved under rather general conditions, such as local stationarity and the existence of derivatives up to the second order of the distribution function. Numerical studies have been carried out to illustrate the performance of our proposals when compared to those procedures requiring the approximation of the indicator variogram. In a final step, the kernel-type estimation of the distribution function has been applied to map the risk of contamination by arsenic in the Central Region of Portugal. With this aim, biomonitoring data of arsenic concentrations were used to detect those zones with higher risk of arsenic accumulation, which is mainly located on the northern part of the region.