Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncated-likelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions. For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.

翻译：给定一个均值为零的高斯随机场，其协方差函数属于某参数族，我们引入一种新的似然近似概念，称为截断似然函数。截断似然函数基于对假定协方差函数族的直接函数近似。针对紧支撑协方差函数，在递增域渐近框架内，我们给出了使基于截断似然函数的估计量保持相合性与渐近正态性的充分条件。我们将该结果应用于广义Wendland协方差函数族，并讨论若干Wendland近似的范例。对于非紧支撑的协方差函数族，我们将结果与协方差尖灭法相结合，证明了当尖灭范围固定时，基于截断-尖灭似然函数的ML估计量在渐近意义上极小化Kullback-Leibler散度。