Estimating spatial quantities under a design-based framework

Spatial statistics has traditionally used the spatial information available before sampling in order to develop efficient sampling designs both under the model- and the design-based framework. Since the paper by de Gruijter and ter Braak we assisted to a revaluation of design-based techniques for inference on spatial data. In more recent papers, Cicchitelli and Montanari and Ghosh et al. , different techniques have been proposed able to take into account the spatial information in design-based inference by adopting a model-assisted estimation of global quantities and by assigning uncertainty to deterministic interpolated spatial surfaces, respectively. However, in our opinion, literature does not present any fully design-based technique able to exploit the spatial information at estimation level. We proposed an estimator for individual quantities coming from a spatial finite population, which uses the geographical information available before sampling by reinterpreting a deterministic interpolator under the design-based framework. Its statistical properties are derived according to the randomness induced by the probabilistic sampling. Moreover, inference for global population quantities is also taken into account as a linear combination of the estimators of individual quantities. In order to better understand the performances of the technique above, we compare it with the simple random sampling estimator in predictive form and with the kriging. The former simply assigns the sampling mean to the unsampled locations, while the latter is a model-based technique widely regarded as the benchmark for inference on spatial data. The performances are computed on the basis of the results of a Monte Carlo experiment where random samples are drawn at different sampling fractions from populations generated according to different spatial configurations. Therefore, it is possible to understand the behavior of the proposed estimator under different settings. The results we obtained highlight that the use of the spatial information at estimation level represents a major boost when compared to the usual design-based techniques. Moreover, the achieved estimator has shown performances not as dissimilar from kriging’s ones as one could have expected.