New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques

Let Z(s)=(Z1(s),…,Zp(s))t be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of Z with the correlation coefficients and measure the spatial dependence with variograms. If two of the Z components are correlated, the spatial information provided by one of them can improve the information of the other. To capture this association, both within components of Z(s) and across s, we use a cross-variogram. Only two robust cross-variogram estimators have been proposed in the literature, both by Lark, and their sample distributions were not obtained. In this paper, we propose new robust cross-variogram estimators, following the location estimation method instead of the scale estimation one considered by Lark, thus extending the results obtained by García-Pérez to the multivariate case. We also obtain accurate approximations for their sample distributions using saddlepoint techniques and assuming a multivariate-scale contaminated normal model. The question of the independence of the transformed variables to avoid the usual dependence of spatial observations is also considered in the paper, linking it with the acceptance of linear variograms and cross-variograms.