On statistical inference for spatial and spatio-temporal extreme processes

Abstract : Natural hazards such as heat waves, extreme wind speeds, and heavy rainfall, arise due to physical processes and are spatial or spatio-temporal in extent. The development of models and inference methods for these processes is a very active area of research. This thesis deals with the statistical inference of extreme and rare events in both spatial and spatio-temporal settings. Specifically, our contributions are dedicated to two classes of stochastic processes: spatial max-mixture processes and space-time max-stable processes. The proposed methodologies are illustrated by applications to rainfall data collected from the East of Australia and from a region in the State of Florida, USA. In the spatial part, we consider hypothesis testing for the mixture parameter a of a spatial maxmixture model using two classical statistics: the Z-test statistic Za and the pairwise likelihood ratio statistic LRa. We compare their performance through an extensive simulation study. The pairwise likelihood is employed for estimation purposes. Overall, the performance of the two statistics is satisfactory. Nevertheless, hypothesis testing presents some difficulties when a lies on the boundary of the parameter space, i.e., a ∈ {0,1}, due to the presence of additional nuisance parameters which are not identified under the null hypotheses. We apply this testing framework in an analysis of exceedances over a large threshold of daily rainfall data from the East of Australia. We also propose a novel estimation procedure to fit spatial max-mixture processes with unknown extremal dependence class. The novelty of this procedure is to provide a way to make inference without specifying the distribution family prior to fitting the data. Hence, letting the data speak for themselves. In particular, the estimation procedure uses nonlinear least squares fit based on a closed form expression of the so-called Fλ-madogram of max-mixture models which contains the parameters of interest. We establish the consistency of the estimator of the mixing parameter a. An indication for asymptotic normality is given numerically. A simulation study shows that the proposed procedure improves empirical coefficients for the class of max-mixture models. In an analysis of monthly maxima of Australian daily rainfall data, we implement the proposed estimation procedure for diagnostic and confirmatory purposes. In the spatio-temporal part, based on a closed form expression of the spatio-temporal Fmadogram, we suggest a semi-parametric estimation methodology for space-time max-stable processes. This part provides a bridge between geostatistics and extreme value theory. In particular, for regular grid observations, the spatio-temporal F-madogram is estimated nonparametrically by its empirical version and a moment-based procedure is applied to obtain parameter estimates. The performance of the method is investigated through an extensive simulation study. Afterward, we apply this method to quantify the extremal behavior of radar daily rainfall maxima data from a region in the State of Florida. This approach could serve as an alternative or a prerequisite to pairwise likelihood estimation. Indeed, the semi-parametric estimates could be used as starting values for the optimization algorithm used to maximize the pairwise log-likelihood function in order to reduce the computational burden and also to improve the statistical efficiency
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Submitted on : Thursday, September 5, 2019 - 9:08:09 AM
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Abdul-Fattah Abu-Awwad. On statistical inference for spatial and spatio-temporal extreme processes. Statistics [math.ST]. Université de Lyon, 2019. English. ⟨NNT : 2019LYSE1079⟩. ⟨tel-02269858⟩

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