Spectral Density Ratio Models for Multivariate Extremes

Carvalho, Miguel de; Davison, A. C.

doi:10.1080/01621459.2013.872651

Cited by 44 publications

(30 citation statements)

References 37 publications

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“…As a result, it seems sensible to determine whether there are any known relationships that can help to make the model parsimonious. An approach suggested by a referee was to adopt a semiparametric specification method similar to that of de Carvalho and Davison (), whereby the different residual densities are interlinked via a tilting term, that is,

\log ((), \frac{{sans-serifg}_{i} false(boldz false)}{{sans-serifg}_{1} false(boldz false)}) = γ_{i} + {boldz}^{T} δ_{i}, for i = 2, …, d

with

{sans-serifg}_{i} false(boldz false) = d G_{i} false(boldz false) false/ d boldz

, where G i is the limiting distribution in expression when conditioning on variable Y i being large, and with ( γ i , δ i ) being constants. If condition holds, the number of parameters reduces to O ( d 2 ).…”

Section: Discussionmentioning

confidence: 99%

Model‐based inference of conditional extreme value distributions with hydrological applications

et al. 2019

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Multivariate extreme value models are used to estimate joint risk in a number of applications, with a particular focus on environmental fields ranging from climatology and hydrology to oceanography and seismic hazards. The semi‐parametric conditional extreme value model of Heffernan and Tawn involving a multivariate regression provides the most suitable of current statistical models in terms of its flexibility to handle a range of extremal dependence classes. However, the standard inference for the joint distribution of the residuals of this model suffers from the curse of dimensionality because, in a d‐dimensional application, it involves a d−1‐dimensional nonparametric density estimator, which requires, for accuracy, a number points and commensurate effort that is exponential in d. Furthermore, it does not allow for any partially missing observations to be included, and a previous proposal to address this is extremely computationally intensive, making its use prohibitive if the proportion of missing data is nontrivial. We propose to replace the d−1‐dimensional nonparametric density estimator with a model‐based copula with univariate marginal densities estimated using kernel methods. This approach provides statistically and computationally efficient estimates whatever the dimension, d, or the degree of missing data. Evidence is presented to show that the benefits of this approach substantially outweigh potential misspecification errors. The methods are illustrated through the analysis of UK river flow data at a network of 46 sites and assessing the rarity of the 2015 floods in North West England.

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\log ((), \frac{{sans-serifg}_{i} false(boldz false)}{{sans-serifg}_{1} false(boldz false)}) = γ_{i} + {boldz}^{T} δ_{i}, for i = 2, …, d

with

{sans-serifg}_{i} false(boldz false) = d G_{i} false(boldz false) false/ d boldz

Section: Discussionmentioning

confidence: 99%

Model‐based inference of conditional extreme value distributions with hydrological applications

et al. 2019

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“…Segers (2009) andde Carvalho & use empirical likelihood to impose the mean constraint (Equation 14) when estimating bivariate extreme-value distributions, and this seems a promising approach that can be broadly applied. Semiparametric models that satisfy the mean constraints have been proposed and fitted using Markov chain Monte Carlo algorithms by Boldi & Davison (2007), Guillotte et al (2011), andSabourin &Naveau (2014).…”

Section: More Complex Settingsmentioning

confidence: 99%

Statistics of Extremes

Davison

Huser

2015

Annu. Rev. Stat. Appl.

159

126

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Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas. 9.1

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“…-define a parametric submodel for either the exponent measure Tawn 1991, 1994) or the spectral measure (Ballani and Schlather 2011;Boldi and Davison 2007); -model in a nonparametric fashion the class of MEVD distributions (Einmahl and Segers 2009;Guillotte et al 2011); -construct models based on other theoretical justifications (De Carvalho and Davison 2014;Ramos and Ledford 2009;Wadsworth et al 2017). In all cases, data are usually transformed via the empirical cdf into Fréchet or uniform margins and then some of the data points, those considered "extreme", are formally retained for inference.…”

Section: Asymptotics and Modelsmentioning

confidence: 99%

Semiparametric bivariate modelling with flexible extremal dependence

Leonelli

Friel

2019

Stat Comput

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Inference over multivariate tails often requires a number of assumptions which may affect the assessment of the extreme dependence structure. Models are usually constructed in such a way that extreme components can either be asymptotically dependent or be independent of each other. Recently, there has been an increasing interest on modelling multivariate extremes more flexibly, by allowing models to bridge both asymptotic dependence regimes. Here we propose a novel semiparametric approach which allows for a variety of dependence patterns, be them extremal or not, by using in a model-based fashion the full dataset. We build on previous work for inference on marginal exceedances over a high, unknown threshold, by combining it with flexible, semiparametric copula specifications to investigate extreme dependence, thus separately modelling marginals and dependence structure. Because of the generality of our approach, bivariate problems are investigated here due to computational challenges, but multivariate extensions are readily available. Empirical results suggest that our approach can provide sound uncertainty statements about the possibility of asymptotic independence, and we propose a criterion to quantify the presence of either extreme regime which performs well in our applications when compared to others available. Estimation of functions of interest for extremes is performed via MCMC algorithms. Attention is also devoted to the prediction of new extreme observations. Our approach is evaluated through simulations, applied to real data and assessed against competing approaches. Evidence demonstrates that the bulk of the data do not bias and improve the inferential process for extremal dependence in our applications.

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Spectral Density Ratio Models for Multivariate Extremes

Cited by 44 publications

References 37 publications

Model‐based inference of conditional extreme value distributions with hydrological applications

Model‐based inference of conditional extreme value distributions with hydrological applications

Statistics of Extremes

Semiparametric bivariate modelling with flexible extremal dependence

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