Convergence in distribution of the L2-deviations of the kernel-type variogram estimators with applications

In this work, some properties of the L2-deviations of the Nadaraya-Watson variogram estimators are analyzed, for both the anisotropic and the isotropic settings. Their convergence in distribution is established, which provides the basis for addressing practical problems, such as the construction of goodness of fit tests for the variogram and, therefore, for modeling the spatial dependence. However, the development of the latter application requires solving different issues, such as the approximation of the model parameters and the critical points. For estimation of the former ones, we propose proceeding through the least squares criteria, whose consistency will be proved, together with a reformulation of the global measures for the kernel-type estimators. Then, the resulting critical points can be approximated by appealing to the bootstrap approaches. Numerical studies with simulated and real data have been developed to illustrate the potentiality of our results, in order to check the appropriateness of a variogram model selected for the variogram.