Estimation of the spatial distribution through the kernel indicator variogram

The distribution function of a spatial random process provides the probability that the variable involved does not exceed a given threshold. Estimation of this function can be addressed in a parametric way, although the available data do not always support the distribution model assumption. Additional options for distribution approximation are based on applying the indicator kriging technique or the sill estimation, which demand assuming stationary conditions from the random process and also estimation of the indicator variogram. In this paper, the latter distribution approximation approaches will be redesigned, by previously establishing the appropriate application setting, so that characterization of the dependence structure will be reduced to the estimation of one indicator variogram for each threshold. Furthermore, a kernel-type estimator is suggested for the latter aim, as a nonparametric alternative to Matheron's indicator variogram. Numerical studies for simulated data have been developed to illustrate the better performance of the kernel estimator, when compared with Matheron's one, as well as the behavior of the procedures described for estimation of the distribution function. Finally, an application is included, where the nitrate concentration in groundwater in the Beja district (Portugal) is measured, so that pollution risk maps of the referred region will have been generated. Copyright © 2012 John Wiley & Sons, Ltd.