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.

空间协方差函数的范围参数估计长期以来一直是空间统计理论和实践问题的根源。特别是，在许多应用中，人们发现似然函数，尤其是受限似然函数，在参数协方差函数模型的范围参数上没有提供任何有意义的上限。这项工作旨在通过表明，至少在某些情况下，只要感兴趣的空间域是有界的，就可以将反范围参数的域扩展到负值，从而进一步深入了解这一现象。这种可能性是通过数值工作、一些有限的理论和对 (Davis, 1973) 高程数据的应用来探索的，这在认识到估计距离参数的困难方面发挥了作用。