**Cari Kaufman, Mark Schervish, and Douglas Nychka**

Maximum likelihood is an attractive method of estimating covariance
parameters in spatial models based on Gaussian processes. However,
calculating the likelihood can be computationally infeasible for large
datasets, requiring O(n^3)
observations. This article proposes the method of covariance tapering
to approximate the likelihood in this setting. In this approach,
covariance matrices are ``tapered,'' or multiplied element-wise by a
sparse correlation matrix. The resulting matrices can then be
manipulated using efficient sparse matrix algorithms. We propose two
approximations to the Gaussian likelihood using tapering. One simply
replaces the model covariance with a tapered version; the other is
motivated by the theory of unbiased estimating equations. Focusing on
the particular case of the Matérn class of covariance functions, we
give conditions under which estimators maximizing the tapering
approximations are, like the maximum likelihood estimator, strongly
consistent. Moreover, we show in a simulation study that the tapering
estimators can have sampling densities quite similar to that of the
maximum likelihood estimate, even when the degree of tapering is
severe. We illustrate the accuracy and computational gains of the
tapering methods in an analysis of yearly total precipitation
anomalies at weather stations in the United States.

*Keywords:* Gaussian process, covariance estimation, compactly supported
correlation function, estimating equations

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