Towards reconciling two asymptotic frameworks in spatial statistics

Two asymptotic frameworks, increasing domain asymptotics and infill asymptotics, have been advanced for obtaining limiting distributions of maximum likelihood estimators of covariance parameters in Gaussian spatial models with or without a nugget effect. These limiting distributions are known to be different in some cases. It is therefore of interest to know, for a given finite sample, which framework is more appropriate. We consider the possibility of making this choice on the basis of how well the limiting distributions obtained under each framework approximate their finite-sample counterparts. We investigate the quality of these approximations both theoretically and empirically, showing that, for certain consistently estimable parameters of exponential covariograms, approximations corresponding to the two frameworks perform about equally well. For those parameters that cannot be estimated consistently, however, the infill asymptotic approximation is preferable.