arXiv:math/0603430v1 [math.ST] 17 Mar 2006 On the Inference of Spartan Spatial Random Field Mod- els for Geostatistical Applications Samuel N. Elogne Department of Mineral Resources Engineering, Technical University of Crete, Chania 73100, GREECE Dionissios T. Hristopulos Department of Mineral Resources Engineering, Technical University of Crete, Chania 73100, GREECE Summary. This paper focuses on the estimation of model parameters (model inference) for the class of Spartan Spatial Random Fields (SSRFs) introduced by Hristopulos (2003). The approach used for model inference involves calculation of sample constraints and fitting with respective ensemble constraints. The fitting leads to optimal SSRF parameters obtained by minimizing a suitable distance functional. We propose kernel-based estimators for calculating the sample constraints from data distributed on irregular sampling grids. We investigate the asymptotic properties of the estimators, and we establish a criterion for the selection of the kernel bandwidth parameters. The performance of the sample constraint estimators, as well as that of the SSRF inference procedure is evaluated by means of numerical simulations for different models of spatial dependence. Keywords: non-parametric, inverse problem, optimization, simulation 1. Introduction During the last 20 years there is an increased interest in spatial random field models (Christakos, 1992; Christakos and Hristopulos, 1998; Yaglom, 1987) and their applications in various scientific disciplines that include statistics, astrophysics, hydrology, ecology, med- ical geography, environmental and petroleum engineering, remote sensing and geographical information systems (GIS). This interest is motivated by the growing availability of spa- tial data and the need for accurate and flexible models of spatial dependence, which allow deriving predictive maps with associated uncertainty estimates. Spatial data are typically sampled on irregular (inhomogeneous) grids (Hall et al., 1994). Geostatistics is a branch of spatial statistics that provides methods for characterizing the spatial dependence and interpolating the data on regular grids. The variogram function is central in these meth- ods. However, inferring the variogram model from the data involves a number of empirical assumptions and it entails, for large sample sizes, heavy computations. A different approach for modelling spatial dependence, based on the Spartan Spatial Random Fields (SSRFs) was recently introduced by Hristopulos (2003). The SSRFs belong in the broad family of Gibbs random fields. SSRFs provide a nonparametric approach for de- termining spatial structure that avoids the subjective choice of dependence structure and the associated regularity conditions (Genton and Gorsich, 2000). In addition, SSRFs present