Showing papers in "Extremes in 2002"
TL;DR: A new class of models is introduced, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions, and who form a newclass of bivariate extreme value distributions.
Abstract: Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.
477 citations
TL;DR: In this article, the authors survey the related asymptotic properties of multivariate distributions, including independence, hidden regular variation and second order regular variation, and discuss the connections and implications.
Abstract: We survey the related asymptotic properties of multivariate distributions; (i) asymptotic independence, (ii) hidden regular variation, and (iii) multivariate second order regular variation. Connections and implications are discussed. The point of view of convergence of measures is emphasized in formulations because we are interested in the concepts being coordinate system free, whenever possible.
200 citations
TL;DR: A new dynamically weighted mixture model, where one term of the mixture is the GPD, and the other is a light-tailed density distribution, which can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.
Abstract: Exceedances over high thresholds are often modeled by fitting a generalized Pareto distribution (GPD) on R+. It is difficult to select the threshold, above which the GPD assumption is enough solid and enough data is available for inference. We suggest a new dynamically weighted mixture model, where one term of the mixture is the GPD, and the other is a light-tailed density distribution. The weight function varies on R+ in such a way that for large values the GPD component is predominant and thus takes the role of threshold selection. The full data set is used for inference on the parameters present in the two component distributions and in the weight function. Maximum likelihood provides estimates with approximate standard deviations. Our approach has been successfully applied to simulated data and to the (previously studied) Danish fire loss data set. We compare the new dynamic mixture method to Dupuis' robust thresholding approach in peaks-over-threshold inference. We discuss robustness with respect to the choice of the light-tailed component and the form of the weight function. We present encouraging simulation results that indicate that the new approach can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.
179 citations
TL;DR: In this paper, the authors deal with the asymptotic and finite sample properties of ρ estimators of the tail index γ, based on external estimators, and show that the behavior of the ρ-estimator considered has a high impact on the distributional properties of the final estimator.
Abstract: In this paper we shall deal with the asymptotic and finite sample properties of “asymptotically unbiased” estimators of the tail index γ, based on “external” adequate estimators of the second order parameter ρ. The behavior of the ρ-estimator considered has indeed a high impact on the distributional properties of the final estimator of γ, and must be carefully chosen. As a by-product of the final study we present also the finite sample properties of a few ρ-estimators available in the literature.
167 citations
TL;DR: In this paper, under quite general conditions, asymptotic justification for the exponential regression model is given as well as for resulting tail index estimators, and diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach are discussed.
Abstract: In Beirlant et al. (1999) and Feuerverger and Hall (1999) an exponential regression model (ERM) was introduced on the basis of scaled log-spacings between subsequent extreme order statistics from a Pareto-type distribution. This lead to the construction of new bias-corrected estimators for the tail index. In this note, under quite general conditions, asymptotic justification for this regression model is given as well as for resulting tail index estimators. Also, we discuss diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach. We show how the diagnostic presented in Guillou and Hall (2001) is linked to the ERM, while a new proposal is suggested. We also provide some small sample comparisons with other existing methods.
114 citations
TL;DR: In this article, a class of semi-parametric estimators for the second order parameter related to a probability distribution with a regularly varying tail is presented, and consistency and asymptotic normality are proven under appropriate conditions.
Abstract: We present a class of semi-parametric estimators for the second order parameter related to a probability distribution with a regularly varying tail. The second order parameter plays an important role whenever dealing with optimization problems in statistics of extreme values. Consistency and asymptotic normality are proven under appropriate conditions.
85 citations
TL;DR: In this paper, the authors introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients, which are the natural dependence measures for multivariate extremal distributions.
Abstract: The extremal coefficients are the natural dependence measures for multivariate extreme value distributions. For an m-variate distribution 2m distinct extremal coefficients of different orders exist; they are closely linked and therefore a complete set of 2m coefficients cannot take any arbitrary values. We give a full characterization of all the sets of extremal coefficients. To this end, we introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients. We construct bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. These bounds are useful as lower order extremal coefficients are the most easily inferred from data.
61 citations
TL;DR: In this paper, asymptotic results for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ℝ n ≥ 0.
Abstract: Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ℝ
N
. They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border ∂ S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.
48 citations
TL;DR: In this article, the problem of estimating the sizes of large inclusions from measurements made on a two-dimensional section of the steel was studied, combining traditional stereological ideas with more recent extreme value modeling.
Abstract: Fatigue properties of steels are strongly influenced by the presence of microscopic particles of oxides or foreign material known as inclusions. The size of the largest inclusion is an important determinant of fatigue strength. This paper studies the problem of estimating the sizes of large inclusions from measurements made on a two-dimensional section of the steel. The approach combines traditional stereological ideas with more recent extreme value modeling. It is shown that both classical likelihood and Bayesian approaches are useful in the inference.
45 citations
TL;DR: In this paper, the authors examined the limiting behavior of (Mn−b(n))/a(n) as n → ∞ for different families of discrete distributions.
Abstract: If X1, X2,..., Xn are independent and identically distributed discrete random variables and Mn=max (X1,..., Xn) we examine the limiting behavior of (Mn−b(n))/a(n) as n → ∞. It is well known that for discrete distributions such as Poisson and geometric the limiting distribution is not non-degenerate. However, by tuning the parameters of the discrete distribution to vary as n → ∞, it is possible to obtain non-degenerate limits for (Mn−b(n))/a(n). We consider four families of discrete distributions and show how this can be done.
36 citations
TL;DR: It is shown that, after scaling, the flooding time TN (p) converges in distribution to the two-fold convolution Λ(2*) of the Gumbel distribution function Λ (z)=exp (−e−z), when the link density pN satisfies NpN/(log N)3 →∞ if N → ∞.
Abstract: Based on our analysis of the hopcount of the shortest path between two arbitrary nodes in the class G
p
(N) of random graphs, the corresponding flooding time is investigated. The flooding time T
N
(p) is the minimum time needed to reach all other nodes from one node. We show that, after scaling, the flooding time T
N
(p) converges in distribution to the two-fold convolution Λ(2*) of the Gumbel distribution function Λ (z)=exp (−e
−z
), when the link density p
N
satisfies Np
N
/(log N)3 → ∞ if N → ∞.
TL;DR: In this paper, the authors apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index, and show that the normal approximation method is worse than the other two methods in terms of coverage probability.
Abstract: For the estimation of the tail index of a heavy tailed distribution, one of the well-known estimators is the Hill estimator (Hill, 1975). One obvious way to construct a confidence interval for the tail index is via the normal approximation of the Hill estimator. In this paper we apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index. Our limited simulation study indicates that the normal approximation method is worse than the other two methods in terms of coverage probability, and the empirical likelihood method and the parametric likelihood method are comparable.
TL;DR: In this article, a measure of pairwise extremal dependence for spatial processes, that is marginally invariant, is introduced, which enables decisions to be made about whether a spatial process is asymptotically dependent or independent for any pair of locations.
Abstract: A measure of pairwise extremal dependence for spatial processes, that is marginally invariant, is introduced. This measure enables decisions to be made about whether a spatial process is asymptotically dependent, asymptotically independent or independent for any pair of locations, thus it provides fundamental diagnostic information for understanding or modeling the extreme values of a spatial process. We illustrate the properties and use of this measure through theoretical examples and applications in hydrology and oceanography.
TL;DR: In this article, the maximum size of random spheres in a reference volume is to be predicted from the size distribution of circles which are planar sections of spheres cut by a plane, if the area of the great circle of spheres have the exponential tail.
Abstract: In the Wicksell corpuscle problem, the maximum size of random spheres in a reference volume is to be predicted from the size distribution of circles which are planar sections of spheres cut by a plane. If the area of the great circle of spheres have the exponential tail, simple prediction methods are applied. Performance of the methods is evaluated by simulation and they are applied to a dataset of graphite nodule sizes in spheroidal graphite cast iron. The effect of left-truncation in Wicksell transform is discussed in a general framework.
TL;DR: In this article, a characterization of strong domains of attraction of joint distributions of a fixed number of extreme generalized order statistics by means of the corresponding result for generalized maxima is given.
Abstract: It is a well-known result in extreme value theory that the von Mises conditions imply the strong convergence of extreme order statistics. We extend this result to extreme generalized order statistics. A characterization of strong domains of attraction of joint distributions of a fixed number of extreme generalized order statistics by means of the corresponding result for generalized maxima is given. In particular, we determine the asymptotic joint distribution of (upper and lower) extreme generalized order statistics. Finally, we show that the Hill estimator based on extreme generalized order statistics is asymptotic normal.
TL;DR: In this paper, an asymptotic evaluation of P{max1≤i≤nXi≤anZ+bn} with Z another Gaussian random vector is obtained for an, bn ∈Rd two vectors obeying certain conditions.
Abstract: Let {Xn, n≥1} be a sequence of independent Gaussian random vectors in Rdd≥2. In this paper an asymptotic evaluation of P{max1≤i≤nXi≤anZ+bn} with Z another Gaussian random vector is obtained for an, bn ∈Rd two vectors obeying certain conditions.
TL;DR: In this paper, it was shown that if the distribution of the maximum of n i.i.d. variables is of the same type for two distinct values of n then the distribution is one of the three extreme value types.
Abstract: In this note, we prove a characterization of extreme value distributions. We show that, under some conditions, if the distribution of the maximum of n i.i.d. variables is of the same type for two distinct values of n then the distribution is one of the three extreme value types. This is an analogue of the well known result that if the sum of two i.i.d. random variables with finite second moment is of the same type as the original distribution then the distribution is Gaussian (Kagan et al., 1973). Our result was motivated by study of the m out of n bootstrap.
TL;DR: In this paper, a formula expressing the inverse cumulative distribution function of a non-negative random variable in terms of contour integrals of its minimal-moment generating function is proved as an analog of the classical continuity theorem for characteristic functions.
Abstract: A formula expressing the inverse cumulative distribution function of a non-negative random variable in terms of contour integrals of its minimal-moment generating function is proved as well as an analog of the classical continuity theorem for characteristic functions.
TL;DR: In this paper, an approximation to the point process of exceedances over a higher threshold is presented, where sharp bounds on the remainder terms when actual distributions of exceedance are replaced by appropriate generalized Pareto distributions (GPD).
Abstract: In this paper we deal with an approximation to the point process of exceedances over a higher threshold. We compute sharp bounds on the remainder terms when actual distributions of exceedances are replaced by appropriate generalized Pareto distributions (GPD). The bound will be formulated in terms of the von Mises function. The ultimate as well as the penultimate approximations are considered; in the latter case the shape parameter of the approximating GPD depends on the threshold.