Local Variogram Models with Negative Inverse “range” Parameters

The estimation of range parameters for spatial covariance functions has long been a source of theoretical and practical problems in spatial statistics. In particular, in many applications, one finds that likelihood and, especially, restricted likelihood functions, do not provide any meaningful upper bound on the range parameter of a parametric covariance function model. This work seeks to provide further insight into this phenomenon by showing that, in at least some circumstances, it can make sense to extend the domain of the inverse range parameter to negative values as long as the spatial domain of interest is bounded. This possibility is explored through numerical work, some limited theory and an application to the (Davis, 1973) elevation data, which played a role in the recognition of the difficulties in estimating range parameters.