Goodness‐of‐fit tests for copula‐based spatial models

There has been a growing interest recently for the modeling of spatial data using multivariate copulas. Such an approach allows for the modeling of spatial dependence independently of the marginal distributions at each site and enables for spatial structures that go beyond the extensively used Gaussian random field. In this context, the choice of an appropriate family of copulas for a given spatial dataset is a crucial issue, in particular when one is interested in accurate spatial interpolations. This paper develops and investigates formal goodness‐of‐fit methodologies for spatial copula models when only one replicate of an isotropic random field is available at a finite number of sites; this setup is standard in geostatistics. Because of the limited information that is available, it is suggested that groups of random pairs sharing similar lag distances be created and that traditional goodness‐of‐fit statistics for bivariate copula families be computed for each group. These statistics are then combined into a global test statistic whose p value is approximated from a suitably adapted parametric bootstrap. The performance of the proposed tests in terms of size and power is investigated in an extensive simulation study. The newly introduced tools are then illustrated on zinc concentration measurements near the Meuse river and on snowfall data in Canada.