Von Mises Conditions Revisited

Falk, Michael; Marohn, Frank

doi:10.1214/aop/1176989120

Cited by 35 publications

(25 citation statements)

References 14 publications

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“…A careful inspection of the proof in the bivariate case shows that the arguments carry over to the d-dimensional case i.e., we have the following result. It generalizes univariate results by Falk and Marohn [10] and Kaufmann [27] to arbitrary dimensions. For related results on the rate of convergence of multivariate extremes in terms of probability metrics we refer to [22,32] as well as the references given there.…”

Section: Spectral D-neighborhoods Of a Gp In R Dsupporting

confidence: 79%

“…We show that the rate of convergence of the maxima in an iid sample of size n is Oðn Àd Þ if and only if the underlying distribution is in the spectral d-neighborhood of a GP. This result extends a bivariate characterization by Falk and Reiss [14] and univariate results by Falk and Marohn [10] and Kaufmann [27] to arbitrary dimensions.…”

supporting

confidence: 84%

“…Hence, GðxÞ is by Lemma 7.2.1 in [35] a df on ðÀN; 0 d : It is obviously max-stable with standard reverse exponential margins and, hence, it coincides with its Pickands representation in (10), which completes the proof. &…”

Section: Domain Of Attractionmentioning

confidence: 57%

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On Pickands coordinates in arbitrary dimensions

Falk¹,

Reiss

2005

Journal of Multivariate Analysis

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Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins.In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral d-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks-over-threshold models are established, which are based on Pickands coordinates. r

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Section: Spectral D-neighborhoods Of a Gp In R Dsupporting

confidence: 79%

supporting

confidence: 84%

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On Pickands coordinates in arbitrary dimensions

Falk¹,

Reiss

2005

Journal of Multivariate Analysis

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“…For G ∈ G with support (a, b), the version of the von Mises condition for the Gumbel case proposed by Falk and Marohn (1993) is (keeping in mind that we use min rather than max)…”

Section: Von Mises Conditionsmentioning

confidence: 99%

Dispersion models for extremes

2009

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We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. A class of extreme generalized linear regression models for analysis of extremes and lifetime data is introduced. The set of quadratic and power slope functions characterize well-known families such as the Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fréchet. We show a convergence theorem for slope functions, by which we may express the classical extreme value convergence results in terms of asymptotics for extreme dispersion models. The key idea is to explore the parallels between location families and natural exponential families, and between the convolution and minimum operations.

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“…In this article we obtain results analogous to those in Falk and Marohn (1993) and Kaufmann (1995). It is shown that the rate of convergence of von Mises type conditions determines the distance of the underlying df F from generalized log Pareto distribution (glogPd, defined later).…”

Section: Introductionmentioning

confidence: 54%