On maximizing the likelihood function of general geostatistical models

Abstract

General geostatistical models are powerful tools for analyzing spatial datasets. A two‐step estimation based on the likelihood function is widely used by researchers, but several theoretical and computational challenges remain to be addressed. First, it is unclear whether there is a unique global maximizer of the log‐likelihood function, a seemingly simple but theoretically challenging question. The second challenge is the convexity of the log‐likelihood function. Besides these two challenges in maximizing the likelihood function, we also study the theoretical property of the two‐step estimation. Unlike many previous works, our results can apply to the non‐twice differentiable covariance functions. In the simulation studies, three optimization algorithms are evaluated in terms of maximizing the log‐likelihood functions.