Abstract:
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The type-G Mat\'ern stochastic partial differential equation (SPDE) random field is an important extension of the well-known SPDE-based formulation of Gaussian random fields to construct non-Gaussian random fields with good theoretical and computational properties. However, unlike Gaussian random fields, whose extremal dependence structure is known in the literature, the extremal dependence structure of the resulted non-Gaussian model has not been studied. Here we show that this exact non-Gaussian random field exhibits only asymptotic independence, whereas its finite element approximation can be both asymptotically independent and asymptotically dependent, depending on whether the closest mesh nodes of the two sites are the same. Moreover, the residual tail dependence coefficient can be computed explicitly in the asymptotic independence case. In terms of statistical modeling, the theoretical results imply that the extremal dependence structure depends on the chosen mesh for the finite element approximation. That is, more accurate approximations require finer meshes, but high-resolution meshes only capture asymptotic dependence at very short distances.
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