  We consider the problem of selecting covariates in spatial linear models with
Gaussian process errors. Penalized maximum likelihood estimation (PMLE) that
enables simultaneous variable selection and parameter estimation is developed
and, for ease of computation, PMLE is approximated by one-step sparse
estimation (OSE). To further improve computational efficiency, particularly
with large sample sizes, we propose penalized maximum covariance-tapered
likelihood estimation (PMLE$_{\mathrm{T}}$) and its one-step sparse estimation
(OSE$_{\mathrm{T}}$). General forms of penalty functions with an emphasis on
smoothly clipped absolute deviation are used for penalized maximum likelihood.
Theoretical properties of PMLE and OSE, as well as their approximations
PMLE$_{\mathrm{T}}$ and OSE$_{\mathrm{T}}$ using covariance tapering, are
derived, including consistency, sparsity, asymptotic normality and the oracle
properties. For covariance tapering, a by-product of our theoretical results is
consistency and asymptotic normality of maximum covariance-tapered likelihood
estimates. Finite-sample properties of the proposed methods are demonstrated in
a simulation study and, for illustration, the methods are applied to analyze
two real data sets.
