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Bachoc, F., Porcu, E., Bevilacqua, M., Furrer, R., & Faouzi, T. (2022). Asymptotically equivalent prediction in multivariate geostatistics. Bernoulli, 28(4), 2518–2545.
Abstract: Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geo-statistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typ-ically multivariate, call for a deeper study of cokriging. In particular, we deal with the problem of misspecified cokriging prediction within the framework of fixed domain asymptotics. Specifically, we provide conditions for equivalence of measures associated with multivariate Gaussian random fields, with index set in a compact set of a d-dimensional Euclidean space. Such conditions have been elusive for over about 50 years of spatial statistics. We then focus on the multivariate Matern and Generalized Wendland classes of matrix valued covariance functions, that have been very popular for having parameters that are crucial to spatial interpolation, and that control the mean square differentiability of the associated Gaussian process. We provide sufficient conditions, for equivalence of Gaussian measures, relying on the covariance parameters of these two classes. This enables to identify the parameters that are crucial to asymptotically equivalent interpolation in multivariate geostatistics. Our findings are then illustrated through simulation studies.
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Garcia-Papani, F., Uribe-Opazo, M. A., Leiva, V., & Aykroyd, R. G. (2017). Birnbaum-Saunders spatial modelling and diagnostics applied to agricultural engineering data. Stoch. Environ. Res. Risk Assess., 31(1), 105–124.
Abstract: Applications of statistical models to describe spatial dependence in geo-referenced data are widespread across many disciplines including the environmental sciences. Most of these applications assume that the data follow a Gaussian distribution. However, in many of them the normality assumption, and even a more general assumption of symmetry, are not appropriate. In non-spatial applications, where the data are uni-modal and positively skewed, the Birnbaum-Saunders (BS) distribution has excelled. This paper proposes a spatial log-linear model based on the BS distribution. Model parameters are estimated using the maximum likelihood method. Local influence diagnostics are derived to assess the sensitivity of the estimators to perturbations in the response variable. As illustration, the proposed model and its diagnostics are used to analyse a real-world agricultural data set, where the spatial variability of phosphorus concentration in the soil is considered-which is extremely important for agricultural management.
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