Nonparametric approaches for direct approximation of the spatial quantiles

The estimation of the spatial quantiles provides information on the thresholds of a spatial variable. This methodology is particularly appealing for its application to data of pollutants, so as to assess their level of risk. A spatial quantile can be approximated through different mechanisms, proposed in the statistics literature, although these approaches suffer from several drawbacks, regarding their lack of optimality or the fact of not leading to direct approximations. Thus, the current work introduces alternative procedures, which try to overcome the aforementioned issues by employing order statistics, similarly as done for independent data. With this aim, the available observations are appropriately transformed to yield a sample of the process at each target site, so that the data obtained are then ordered and used to derive the spatial quantile at the corresponding location. The new methodology can be directly applied to data from processes that are either stationary or that deviate from this condition for a non-constant trend and, additionally, it can be even extended to heteroscedastic data. Simulation studies under different scenarios have been accomplished, whose results show the adequate performance of the proposed estimators. A further step of this research is the application of the new approaches to data of nitrogen dioxide concentrations, to exemplify the potential of the quantile estimates to check the thresholds of a pollutant at a specific moment, as well as their evolution over time.