Contribution à la modélisation spatiale des événements extrêmes (original) (raw)
Related papers
Estimation of spatial max-stable models using threshold exceedances
Statistics and Computing, 2013
Parametric inference for spatial max-stable processes is difficult since the related likelihoods are unavailable. A composite likelihood approach based on the bivariate distribution of block maxima has been recently proposed. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block maxima, typically annual, is unable to take into account the fact that maxima occur or not simultaneously. If time series of, say, daily data are available, then estimation procedures based on exceedances of a high threshold could mitigate such problems. We focus on two approaches for composing likelihoods based on pairs of exceedances. The first one comes from the tail approximation for bivariate distribution proposed by Ledford and Tawn (Biometrika 83:169-187, 1996) when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension (Rootzén and Tajvidi in Bernoulli 12: [917][918][919][920][921][922][923][924][925][926][927][928][929][930] 2006) of the generalized Pareto distribution which allows to model exceedances when at least one of the components is over the threshold. The two approaches are compared through a simulation study where both processes in a domain of attraction of a max-stable process and max-stable processes are successively considered as time replications, according to different degrees of spatial dependency. Results put forward how the nature of the time replications influences the bias of estimations and highlight the choice J.-N. Bacro ( ) I3M, Université Montpellier II, 4, Place Eugène Bataillon, of each approach regarding to the strength of the spatial dependencies and the threshold choice.
Threshold selection for extremes under a semiparametric model
Statistical Methods & Applications, 2013
In this work we propose a semiparametric likelihood procedure for the threshold selection for extreme values. This is achieved under a semiparametric model, which assumes there is a threshold above which the excess distribution belongs to the generalized Pareto family. The motivation of our proposal lays on a particular characterization of the threshold under the aforementioned model. A simulation study is performed to show empirically the properties of the proposal and we also compare it with other estimators.
Asymptotic models and inference for extremes of spatio-temporal data
Extremes, 2010
Recently there has been a lot of effort to model extremes of spatially dependent data. These efforts seem to be divided into two distinct groups: the study of max-stable processes, together with the development of statistical models within this framework; the use of more pragmatic, flexible models using Bayesian hierarchical models (BHM) and simulation based inference techniques. Each modeling strategy
Estimation of extreme Pareto quantiles using upper order statistics
2002
A common feature of certain hydrological data plotted vs their return period on a log-log scale is an apparent straight line fit at the upper end. This is compatible with an assumed two-parameter Pareto model for the upper end. The objective of this study is to utilize only the largest sample observations in the estimation of extreme upper quantiles. Parameters of the Pareto are estimated based on the upper order statistics at a single site using maximum likelihood. Also, regional estimates are obtained under an assumed indexing method that yields a type of regional homogeneity. Exact formulas for the mean squared error of quantile estimators are given. Extreme precipitation data from the northern front range of Colorado, USA, are used for illustrating the procedures.
A Bayesian semi-parametric hybrid model for spatial extremes with unknown dependence structure
2020
The max-stable process is an asymptotically justified model for spatial extremes. In particular, we focus on the hierarchical extreme-value process (HEVP), which is a particular max-stable process that is conducive to Bayesian computing. The HEVP and all max-stable process models are parametric and impose strong assumptions including that all marginal distributions belong to the generalized extreme value family and that nearby sites are asymptotically dependent. We generalize the HEVP by relaxing these assumptions to provide a wider class of marginal distributions via a Dirichlet process prior for the spatial random effects distribution. In addition, we present a hybrid max-mixture model that combines the strengths of the parametric and semi-parametric models. We show that this versatile max-mixture model accommodates both asymptotic independence and dependence and can be fit using standard Markov chain Monte Carlo algorithms. The utility of our model is evaluated in Monte Carlo sim...
From generalized Pareto to extreme values law: Scaling properties and derived features
Journal of Geophysical Research, 2001
Given the fact that, assuming a generalized Pareto distribution for a process, it is possible to derive an asymptotic generalized extreme values law for the corresponding maxima, in this paper we consider the theoretical relations linking the parameters of such distributions. In addition, temporal scaling properties are shown to hold for both laws when considering proper power-law forms for both the position and the scale parameters; also shown is the relation between the scaling exponents of the distributions of interest, how the scaling properties of one distribution yield those of the other, and how the scaling features may be used to estimate the parameters of the distributions at different temporal scales. Finally, an application to rainfall is given.
The Annals of Statistics, 2021
Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit non-parametric estimation and introduce rank-based Mestimators for parametric models that simultaneously work under asymptotic dependence and asymptotic independence, without requiring prior knowledge on which of the two regimes applies. Asymptotic normality of the proposed estimators is established under weak regularity conditions. We further show how bivariate estimators can be leveraged to obtain parametric estimators in spatial tail models, and again provide a thorough theoretical justification for our approach.
A survey of spatial extremes: Measuring spatial dependence and modeling spatial effects
2012
We survey the current practice of analyzing spatial extreme data, which lies at the intersection of extreme value theory and geostatistics. Characterizations of multivariate max-stable distributions typically assume specific univariate marginal distributions, and their statistical applications generally require capturing the tail behavior of the margins and describing the tail dependence among the components. We review current methodology for spatial extremes analysis, discuss the extension of the finite-dimensional extremes framework to spatial processes, review spatial dependence metrics for extremes, survey current modeling practice for the task of modeling marginal distributions, and then examine max-stable process models and copula approaches for modeling residual spatial dependence after accounting for marginal effects.