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On estimation in random fields generated by linear stochastic partial differential equations. (English) Zbl 0940.62089
Linear stochastic partial differential equations of the form \[ (1)\qquad \mathcal P_\theta (\partial)c=\sigma \varepsilon \] are studied, where \(\mathcal P_\theta (\partial)\) is a formal linear partial differential operator, \(c\) and \(\varepsilon \) are generalized random fields, or, equivalently, distribution-valued processes, and \(\theta \), \(\sigma \) are unknown parameters. This paper explains the mathematical meaning of (1) as well as the relations between the solutions of (1) and the random field models used in statistics. Also, the relations between the solutions of (1) and their applicability to the analysis of spatial data are discussed, in particular to parameter estimation. Several nice examples are given for illustration of the theory.
MSC:
62M30 Inference from spatial processes
62M40 Random fields; image analysis
60G60 Random fields
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