  Maximizing the likelihood has been widely used for estimating the unknown
covariance parameters of spatial Gaussian processes. However, evaluating and
optimizing the likelihood function can be computationally intractable,
particularly for large number of (possibly) irregularly spaced observations,
due to the need to handle the inverse of ill-conditioned and large covariance
matrices. Extending the "inversion-free" method of Anitescu, Chen and Stein
\cite{anitescu2017inversion}, we investigate a broad class of covariance
parameter estimators based on inversion-free surrogate losses and block
diagonal approximation schemes of the covariance structure. This class of
estimators yields a spectrum for negotiating the trade-off between statistical
accuracy and computational cost. We present fixed-domain asymptotic properties
of our proposed method, establishing $\sqrt{n}$-consistency and asymptotic
normality results for isotropic Matern Gaussian processes observed on a
multi-dimensional and irregular lattice. Simulation studies are also presented
for assessing the scalability and statistical efficiency of the proposed
algorithm for large data sets.
