Maxima of normal random vectors: Between independence and complete dependence

Hüsler, Jürg; Reiss, Rolf–Dieter

doi:10.1016/0167-7152(89)90106-5

Cited by 248 publications

(271 citation statements)

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“…In this case expression (1) boils down to the Hüsler-Reiss (1989) model for bivariate extremes. The bivariate marginal density functions f (z 1 , z 2 ) are easily expressed using derivatives of (1).…”

Section: Introductionmentioning

confidence: 99%

Composite likelihood estimation for the Brown-Resnick process

Huser¹,

Davison²

2013

Biometrika

109

105

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SUMMARY Genton et al. (2011) investigated the gain in efficiency when triplewise, rather than pairwise, likelihood is used to fit the popular Smith max-stable model for spatial extremes. We generalize their results to the Brown-Resnick model and show that the efficiency gain is substantial only for very smooth processes, which are generally unrealistic in applications.Some key words: Brown-Resnick process; Composite likelihood; Max-stable process; Pairwise likelihood; Smith process; Triplewise likelihood. INTRODUCTIONMax-stable processes are useful for the statistical modelling of spatial extreme events. No finite parametrization of such processes exists, but a spectral representation (de Haan, 1984) aids in constructing models. In a 1990 University of Surrey technical report, R. L. Smith proposed a max-stable model based on deterministic storm profiles which has become popular because it is simple, readily interpreted and easily simulated, but unfortunately it is too inflexible for realism in practice. Another popular model, the Brown-Resnick process, is based on intrinsically stationary log-Gaussian processes, can handle a wide range of dependence structures, and often provides a better fit to data; see, for example, or a 2012 University of North Carolina at Chapel Hill PhD thesis by Soyoung Jeon. Kabluchko et al. (2009) provided further underpinning for this process by showing that that under mild conditions, the BrownResnick process with variogram 2γ(h) = ( h /ρ) α (ρ > 0, 0 < α ≤ 2), where h is the spatial lag, is essentially the only isotropic limit of properly rescaled maxima of Gaussian processes. The Smith model is obtained by taking a Brown-Resnick process with variogram 2γ(h) = h T Σ −1 h for some covariance matrix Σ, corresponding after an affine transformation to taking α = 2, whereas found that 1/2 < α < 1 for the rainfall data they examined.Likelihood inference for max-stable models is difficult, since only the bivariate marginal density functions are known in most cases, and pairwise marginal likelihood is typically used (Padoan et al., 2010;Davison and Gholamrezaee, 2012;Huser and Davison, 2012). This raises the question whether some other approach to inference would be preferable. Genton et al. (2011) derived the general form of the likelihood function for the Smith model and showed that large efficiency gains can arise when fitting it using triplewise, rather than pairwise, likelihood. In this paper we extend their investigation to the Brown-Resnick process and show that for rougher models, more realistic than those considered by Genton et al. (2011), the efficiency gains are much less striking. Thus pairwise likelihood inference provides a good compromise between statistical and computational efficiency in many applications.

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Section: Introductionmentioning

confidence: 99%

Composite likelihood estimation for the Brown-Resnick process

Huser¹,

Davison²

2013

Biometrika

109

105

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“…Implemented bivariate extreme value copulas are Galambos copula (Galambos 1987) and Husler-Reiss copula (Hüsler and Reiss 1989), in addition to Gumbel copula (Gumbel 1960) which is also an Archimedean copula. Implemented bivariate association measures are Kendall's tau, Spearman's rho, and tail index (lower and upper).…”

Section: Discussionmentioning

confidence: 99%

Enjoy the Joy of Copulas: With a Packagecopula

Yan¹

2007

J. Stat. Soft.

387

212

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Copulas have become a popular tool in multivariate modeling successfully applied in many fields. A good open-source implementation of copulas is much needed for more practitioners to enjoy the joy of copulas. This article presents the design, features, and some implementation details of the R package copula. The package provides a carefully designed and easily extensible platform for multivariate modeling with copulas in R. S4 classes for most frequently used elliptical copulas and Archimedean copulas are implemented, with methods for density/distribution evaluation, random number generation, and graphical display. Fitting copula-based models with maximum likelihood method is provided as template examples. With the classes and methods in the package, the package can be easily extended by user-defined copulas and margins to solve problems.

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“…In the finitedimensional setting, the asymptotic behavior of the maxima of Gaussian random vectors was first investigated by Huesler and Reiss [13]. Recently, Kabluchko, Schlather and de Haan [15] and Kabluchko [14] consider the functional setting and prove convergence of the maxima of i.i.d.…”

Section: Gaussian Casementioning

confidence: 99%

Extremes of independent stochastic processes: a point process approach

Éyi-Minko

Dombry

2016

Extremes

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For each n ≥ 1, let {X in , i1} be independent copies of a nonnegative continuous stochastic process X n = (X n (t)) t∈T indexed by a compact metric space T . We are interested in the process of partial maximãwhere the brackets [ · ] denote the integer part. Under a regular variation condition on the sequence of processes X n , we prove that the partial maxima processM n weakly converges to a superextremal processM as n → ∞. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

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Maxima of normal random vectors: Between independence and complete dependence

Cited by 248 publications

References 1 publication

Composite likelihood estimation for the Brown-Resnick process

Composite likelihood estimation for the Brown-Resnick process

Enjoy the Joy of Copulas: With a Packagecopula

Extremes of independent stochastic processes: a point process approach

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