We consider the problem of selecting covariates in spatial lin- ear models with Gaussian process errors. Penalized maximum likeli- hood 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 fur- ther improve computational efficiency particularly with large sample sizes, we propose penalized maximum covariance-tapered likelihood estimation (PMLE_T) and its one-step sparse estimation (OSE_T). General forms of penalty functions with an emphasis on smoothly clipped absolute deviation are used for penalized maximum likeli- hood. Theoretical properties of PMLE and OSE, as well as their approximations PMLET and OSET using covariance tapering are de- rived, including consistency, sparsity, asymptotic normality, and the oracle properties. For covariance tapering, a by-product of our theo- retical 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.
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