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21
Bayesian inference from composite likelihoods, with an application to spatial extremes. Statistica Sinica 22: 813–845
, 2012
"... Composite likelihoods are increasingly used in applications where the full likelihood is analytically unknown or computationally prohibitive. Although some frequentist properties of the maximum composite likelihood estimator are akin to those of the maximum likelihood estimator, Bayesian inference b ..."
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Composite likelihoods are increasingly used in applications where the full likelihood is analytically unknown or computationally prohibitive. Although some frequentist properties of the maximum composite likelihood estimator are akin to those of the maximum likelihood estimator, Bayesian inference based on composite likelihoods is in its early stages. This paper discusses inference when one uses composite likelihood in Bayes ’ formula. We establish that using a composite likelihood results in a proper posterior density, though it can differ considerably from that stemming from the full likelihood. Building on previous work on composite likelihood ratio tests, we use asymptotic theory for misspecified models to propose two adjustments to the composite likelihood to obtain appropriate inference. We also investigate use of the Metropolis Hastings algorithm and two implementations of the Gibbs sampler for obtaining draws from the composite posterior. We test the methods on simulated data and apply them to a spatial extreme rainfall dataset. For the simulated data, we find that posterior credible intervals yield appropriate empirical coverage rates. For the extreme precipitation data, we are able to both effectively model marginal behavior throughout the study region and obtain appropriate measures of spatial dependence.
Spacetime modelling of extreme events
 Journal of the Royal Statistical Society: Series B (Statistical Methodology
, 2014
"... Maxstable processes are the natural analogues of the generalized extremevalue distribution for the modelling of extreme events in space and time. Under suitable conditions, these processes are asymptotically justified models for maxima of independent replications of random fields, and they are als ..."
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Maxstable processes are the natural analogues of the generalized extremevalue distribution for the modelling of extreme events in space and time. Under suitable conditions, these processes are asymptotically justified models for maxima of independent replications of random fields, and they are also suitable for the modelling of joint individual extreme measurements over high thresholds. This paper extends a model of Schlather (2002) to the spacetime framework, and shows how a pairwise censored likelihood can be used for consistent estimation under mild mixing conditions. Estimator efficiency is also assessed and the choice of pairs to be included in the pairwise likelihood is discussed based on computations for simple time series models. The ideas are illustrated by an application to hourly precipitation data over Switzerland.
Estimation of Hüsler–Reiss distributions and Brown–Resnick processes
, 2012
"... Summary. Estimation of extremevalue parameters from observations in the maxdomain of attraction (MDA) of a multivariate maxstable distribution commonly uses aggregated data such as block maxima. Since we expect that additional information is contained in the nonaggregated, single “large ” observ ..."
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Cited by 9 (1 self)
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Summary. Estimation of extremevalue parameters from observations in the maxdomain of attraction (MDA) of a multivariate maxstable distribution commonly uses aggregated data such as block maxima. Since we expect that additional information is contained in the nonaggregated, single “large ” observations, we introduce a new approach of inference based on a multivariate peaksoverthreshold method. We show that for any process in the MDA of the frequently used HüslerReiss model or its spatial extension, the BrownResnick process, suitably defined conditional increments asymptotically follow a multivariate Gaussian distribution. This leads to computationally efficient estimates of the HüslerReiss parameter matrix. Further, the results enable parametric inference for BrownResnick processes. A simulation study compares the performance of the new estimators to other commonly used methods. As an application, we fit a nonisotropic BrownResnick process to the extremes of 12 year data of daily wind speed measurements.
Approximate Bayesian computing for spatial extremes
 Computational Statistics & Data Analysis
, 2012
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A survey of spatial extremes: Measuring spatial dependence and modeling spatial effects
 Revstat
, 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 maxstable distributions typically assume specific univariate marginal distributions, and their statistical applications gener ..."
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We survey the current practice of analyzing spatial extreme data, which lies at the intersection of extreme value theory and geostatistics. Characterizations of multivariate maxstable 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 finitedimensional 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 maxstable process models and copula approaches for modeling residual spatial dependence after accounting for marginal effects. KeyWords: copula; extremal coefficient; hierarchical model; madogram; maxstable process; multivariate extreme value distribution. AMS Subject Classification: • 62M30, 62H11, 62H20. 136 D. Cooley et al.A Survey of Spatial Extremes 137
Estimation of spatial maxstable models using threshold exceedances. (Available from http://arxiv.org/abs/1205.1107
, 2012
"... Parametric inference for spatial maxstable 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 in the literature. However modeling block maxima is a wasteful approach pro ..."
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Cited by 4 (0 self)
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Parametric inference for spatial maxstable 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 in the literature. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block, typically annual, maxima 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. In this paper 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 (1996) when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension (Rootzén and Tajvidi, 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 according to different degrees of spatial dependency. Results show that both the strength of the spatial dependencies and the threshold choice play a fundamental role in determining which is the best estimating procedure.
Efficient inference and simulation for elliptical Pareto processes
, 2014
"... Recent advances in spatial extreme value theory have established Pareto processes as the natural limits for threshold exceedances of spatial processes. `Pareto processes are obtained by considering exceedances of a risk functional `, defined as for instance the spatial supremum. Here we introduce e ..."
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Recent advances in spatial extreme value theory have established Pareto processes as the natural limits for threshold exceedances of spatial processes. `Pareto processes are obtained by considering exceedances of a risk functional `, defined as for instance the spatial supremum. Here we introduce elliptical `Pareto processes, which arise as the limit of threshold exceedances of certain elliptical processes and provide a flexible dependence model characterized by a correlation function and a shape parameter. These processes correspond to extremalt limit processes for rescaled maxima. We introduce an efficient inference method for them based on maximizing a full likelihood with partial censoring of components falling below a high marginal threshold and we develop exact conditional and unconditional simulation algorithms. These ideas are illustrated by modelling precipitation extremes in a region of Switzerland.
DOI: 10.1002/env.000 Spatial Extreme Value Analysis to Project Extremes of LargeScale Indicators for Severe Weather
, 2013
"... Summary: Extreme weather is of great concern under a changing climate because of disproportional impacts on society. Most such events occur at scales that are too fine for global (or even most regional) climate models to resolve, making it difficult to infer potential future changes and impacts. One ..."
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Summary: Extreme weather is of great concern under a changing climate because of disproportional impacts on society. Most such events occur at scales that are too fine for global (or even most regional) climate models to resolve, making it difficult to infer potential future changes and impacts. One approach to analyzing severe weather in future climates is to consider largerscale processes that are providing the setting for finer scale events. Concurrently high values of convective available potential energy (CAPE) and 06 km wind shear (Shear) have been found to represent conducive environments for severe weather. Here, we analyze these variables to determine how they might be used to project the spatial context of environments conducive to severe weather. Specific challenges include the fact that the data have strong spatial dependences, and analyzing extreme values over space is an active area of research. We take a new approach based on the Heffernan and Tawn conditional extreme value model. Results suggest that this technique excels at estimating the spatial behavior of CAPE and Shear largely because it allows for modeling their entire distribution, not just the extremes. A case study further examines these variables conditional on high river flow events, and it is found that distinct spatial patterns in the largescale variables tend to exist concurrently with high river flow.
Mathematical Geosciences manuscript No. (will be inserted by the editor) Geostatistics of Dependent and Asymptotically Independent Extremes
"... Abstract Spatial modeling of rare events has obvious applications in the environmental sciences and is crucial when assessing the effects of catastrophic events, such as heatwaves or widespread flooding, on food security and on the sustainability of societal infrastructure. Although classical geosta ..."
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Abstract Spatial modeling of rare events has obvious applications in the environmental sciences and is crucial when assessing the effects of catastrophic events, such as heatwaves or widespread flooding, on food security and on the sustainability of societal infrastructure. Although classical geostatistics is largely based on Gaussian processes and distributions, these are not appropriate for extremes, for which maxstable and related processes provide more suitable models. This paper provides a brief overview of current work on the statistics of spatial extremes, with an emphasis on the consequences of the assumption of maxstability. Applications to winter minimum temperatures and daily rainfall are described.