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Parametric and semiparametric model estimation and selection in geostatistics




Chu, Tingjin, author
Wang, Haonan, advisor
Zhu, Jun, advisor
Meyer, Mary, committee member
Luo, J. Rockey, committee member

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This dissertation is focused on geostatistical models, which are useful in many scientific disciplines, such as climatology, ecology and environmental monitoring. In the first part, we consider variable selection 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 (PMLET) and its one-step sparse estimation (OSET). 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 PMLET and OSET 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. In the second part, we develop a new semiparametric approach to geostatistical modeling and inference. In particular, we consider a geostatistical model with additive components, where the covariance function of the spatial random error is not pre-specified and thus flexible. A novel, local Karhunen-Loève expansion is developed and a likelihood-based method devised for estimating the model parameters. In addition, statistical inference, including spatial interpolation and variable selection, is considered. Our proposed computational algorithm utilizes Newton-Raphson on a Stiefel manifold and is computationally efficient. A simulation study demonstrates sound finite-sample properties and a real data example is given to illustrate our method. While the numerical results are comparable to maximum likelihood estimation under the true model, our method is shown to be more robust against model misspecification and is computationally far more efficient for larger sample sizes. Finally, the theoretical properties of the estimates are explored and in particular, a consistency result is established.


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