The change-of-variance function: A tool to explore the effects of dependencies in spatial statistics

This paper presents the computation of the change-of-variance function of M-estimators of scale under general contamination for dependent observations. In this context, several results of robustness are established, and the links between B-robustness, V-robustness and V^◇-robustness are studied. Some more specific properties are derived for Gaussian distributions. These results are then applied to variogram estimation, which is a crucial stage of spatial prediction. The change-of-variance function is shown to be a tool to explore the effects of dependencies on the variance of variogram estimators. ARMA models are used in order to model unidirectional spatial dependencies. It is shown that the shape of the change-of-variance function under dependence is characteristic of the type of variogram estimator. However, this shape depends also on the underlying dependency structure, its intensity, as well as the lag distance of the variogram estimates. Therefore, statistical insight is provided into the sensitivity and the behavior of the variance of the variogram estimator at different spatial lags. For instance, this variance plays an important role when fitting a parametric variogram model by weighted or generalized least squares.