Estimating spatial quantities under a design-based framework Abstract

ITACOSM 2013Vagheggini, A. 1 , Bruno, F. 2 and Cocchi D. 2Estimating spatial quantities under a design-based frameworkAbstractSpatial statistics has traditionally used the spatial information available before sampling in order todevelop efficient sampling designs both under the model- and the design-based framework. Sincethe paper by de Gruijter and ter Braak [3] we assisted to a revaluation of design-based techniquesfor inference on spatial data. In more recent papers, Cicchitelli and Montanari [2] and Ghosh et al.[4], different techniques have been proposed able to take into account the spatial information indesign-based inference by adopting a model-assisted estimation of global quantities and byassigning uncertainty to deterministic interpolated spatial surfaces, respectively. However, in ouropinion, literature does not present any fully design-based technique able to exploit the spatialinformation at estimation level. We proposed an estimator for individual quantities coming from aspatial finite population, which uses the geographical information available before sampling byreinterpreting a deterministic interpolator under the design-based framework [1]. Its statisticalproperties are derived according to the randomness induced by the probabilistic sampling.Moreover, inference for global population quantities is also taken into account as a linearcombination of the estimators of individual quantities [5].In order to better understand the performances of the technique above, we compare it with thesimple random sampling estimator in predictive form and with the kriging. The former simplyassigns the sampling mean to the unsampled locations, while the latter is a model-based techniquewidely regarded as the benchmark for inference on spatial data. The performances are computed onthe basis of the results of a Monte Carlo experiment where random samples are drawn at differentsampling fractions from populations generated according to different spatial configurations.Therefore, it is possible to understand the behavior of the proposed estimator under differentsettings. The results we obtained highlight that the use of the spatial information at estimation levelrepresents a major boost when compared to the usual design-based techniques. Moreover, theachieved estimator has shown performances not as dissimilar from kriging’s ones as one could haveexpected.References[1] Bruno, F., Cocchi, D. and Vagheggini, A. (2012) Finite population properties of individualpredictors based on spatial patterns, Environ. Ecol. Stat., advance online publication.[2] Cicchitelli, G. and Montanari, G.E. (2012) Model-assisted estimation of a spatial populationmean, Int. Stat. Rev. 00, 1_16.[3] de Gruijter, J. and ter Braak, C. (1990) Model-free estimation from spatial samples: areappraisal of classical sampling theory, Math. Geol. 22, 407_415.1. Department of Statistical Sciences, University of Bologna, Italy, alessandr.vagheggin2@unibo.it2. Department of Statistical Sciences, University of Bologna, Italy, francesca.bruno@unibo.it;daniela.cocchi@unibo.it

ITACOSM 2013[4] Ghosh, S., Gelfand, A.E. and Mølhave, T. (2012) Attaching uncertainty to deterministic spatialinterpolations, Stat. Methodology 9, 251_264.[5] Vagheggini, A. (2013) Employing Distances in Design-based Spatial Estimation, Ph.D.dissertation, University of Bologna, Italy.1. Department of Statistical Sciences, University of Bologna, Italy, alessandr.vagheggin2@unibo.it2. Department of Statistical Sciences, University of Bologna, Italy, francesca.bruno@unibo.it;daniela.cocchi@unibo.it