Showing papers in "Extremes in 2008"
TL;DR: In the last decades, there has been a shift from the parametric statistics of extremes for IID random variables, based on the probabilistic asymptotic results in extreme value theory, towards a semi-parametric approach, where the estimation of the right tail-weight, under a quite general framework, is of major importance as discussed by the authors.
Abstract: In the last decades there has been a shift from the parametric statistics of extremes for IID random variables, based on the probabilistic asymptotic results in extreme value theory, towards a semi-parametric approach, where the estimation of the right tail-weight, under a quite general framework, is of major importance. After a brief presentation of classical Gumbel’s block methodology and of later improvements in the parametric framework (multivariate and multi-dimensional extreme value models for largest observations and peaks over threshold approaches), we present a coordinated overview, over the last three decades, of the developments on the estimation of the extreme value index under a semiparametric framework. Laurens de Haan has been one of the leading scientists in the field, (co-)author of many seminal ideas, that he generously shared with dozens (literally) of colleagues and students, thus achieving one of the main goals in a scientist’s life: he gathered around him a bunch of colleagues united in the endeavour of building knowledge. The last section is a personal tribute to Laurens, who fully lives his ideal that co-operation is the heart of Science.
63 citations
TL;DR: In this article, it was shown that max-stable random vectors with unit Frechet marginals are in one to one correspondence with convex sets K in [0, ∞ )====== d called max-zonoids, which can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or as the selection expectation of a random crosspolytope whose distribution is controlled by the spectral measure of the max-safe vector.
Abstract: It is shown that max-stable random vectors in [0, ∞ )
d
with unit Frechet marginals are in one to one correspondence with convex sets K in [0, ∞ )
d
called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function P
ξ ≤ x of a max-stable random vector ξ with unit Frechet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided.
58 citations
TL;DR: In this article, it was shown that the distribution of the multiplier Y belongs to the class L(γ) or S(γ ) for some γ ≥ 0 and Y is unbounded.
Abstract: Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class L(γ ) or S(γ ) for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class L(0) or S(0) accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.
53 citations
TL;DR: In this article, the authors considered the random sums of i.i.d. random variables with consistent variation and obtained the asymptotic behavior of the tail P(ξ1 + ξ1+ξ 1+ν 2 + ν 3+Ω 4 + Ω 5 + ε 2+ε 4 +ε 5 +Ω 6+δ 7 +δ 8 +ξ η.
Abstract: In this paper, we consider the random sums of i.i.d. random variables ξ
1,ξ
2,... with consistent variation. Asymptotic behavior of the tail P(ξ1 + ... + ξη > x), where η is independent of ξ
1,ξ
2,..., is obtained for different cases of the interrelationships between the tails of ξ
1 and η. Applications to the asymptotic behavior of the finite-time ruin probability ψ(x,t) in a compound renewal risk model, earlier introduced by Tang et al. (Stat Probab Lett 52, 91–100 (2001)), are given. The asymptotic relations, as initial capital x increases, hold uniformly for t in a corresponding region. These asymptotic results are illustrated in several examples.
41 citations
TL;DR: In this article, the authors review some novel applications of the bootstrap and the empirical likelihood techniques in extreme-value statistics, including estimating the optimal sample fractions and constructing confidence intervals of various quantities.
Abstract: One of the major interests in extreme-value statistics is to infer the tail properties of the distribution functions in the domain of attraction of an extreme-value distribution and to predict rare events. In recent years, much effort in developing new methodologies has been made by many researchers in this area so as to diminish the impact of the bias in the estimation and achieve some asymptotic optimality in inference problems such as estimating the optimal sample fractions and constructing confidence intervals of various quantities. In particular, bootstrap and empirical likelihood methods, which have been widely used in many areas of statistics, have drawn attention. This paper reviews some novel applications of the bootstrap and the empirical likelihood techniques in extreme-value statistics.
40 citations
TL;DR: In this article, a review of testing issues in extreme value theory is presented, which includes research done by Professor Laurens de Haan and many others, and some practical questions in this direction.
Abstract: As a leading statistician in extreme value theory, Professor Laurens de Haan has made significant contribution in both probability and statistics of extremes. In honor of his 70th birthday, we review testing issues in extremes, which include research done by Professor Laurens de Haan and many others. In comparison with statistical estimation in extremes, research on testing has received less attention. So we also point out some practical questions in this direction.
23 citations
TL;DR: In this paper, the authors discuss two aspects of the statistical inference on the extreme value behavior of time series with a particular emphasis on his important contributions, and compare the performance of a direct marginal tail analysis with that of a model-based approach using an analysis of residuals.
Abstract: On the occasion of Laurens de Haan’s 70th birthday, we discuss two aspects of the statistical inference on the extreme value behavior of time series with a particular emphasis on his important contributions. First, the performance of a direct marginal tail analysis is compared with that of a model-based approach using an analysis of residuals. Second, the importance of the extremal index as a measure of the serial extremal dependence is discussed by the example of solutions of a stochastic recurrence equation.
19 citations
TL;DR: An inference procedure based on an automatic declustering scheme is developed, and using simulated data, this procedure is implemented and assessed, making inferences for the extremal index, and for two cluster functionals.
Abstract: We consider Bayesian inference for the extremes of dependent stationary series. We discuss the virtues of the Bayesian approach to inference for the extremal index, and for related characteristics of clustering behaviour. We develop an inference procedure based on an automatic declustering scheme, and using simulated data we implement and assess this procedure, making inferences for the extremal index, and for two cluster functionals. We then apply our procedure to a set of real data, specifically a time series of wind-speed measurements, where the clusters correspond to storms. Here the two cluster functionals selected previously correspond to the mean storm length and the mean inter-storm interval. We also consider inference for long-period return levels, advocating the posterior predictive distribution as being most representative of the information required by engineers interested in design level specifications.
19 citations
TL;DR: In this paper, the authors study the extremal behavior of a daily maximum sea water levels series, which is presented in Draisma (Duration of extremes at sea), and obtain the limit behavior of the tail empirical quantile function associated with a random sample and hence the asymptotic normality of a class of estimators for the tail index that includes Hill estimator.
Abstract: This work emerges from a study of the extremal behavior of a daily maximum sea water levels series, {X
i
} , presented in Draisma (Duration of extremes at sea. In: Parametric and semi-parametric methods in E. V. T., pp. 137–143. PhD thesis, Erasmus, University, 2001). In its approach, a new series, {Y
i
}, is defined, consisting of water levels that persist for a fixed period of time. In this paper, we study the tail behavior of {Y
i
} , in case {X
i
} is independent and identically distributed (i.i.d.) and in case {X
i
} is a max-autoregressive sequence (we will consider two different max-autoregressive processes), whose distribution function is in the Frechet domain of attraction. We also determine Ledford and Tawn tail dependence index (Ledford and Tawn, Biometrika 83:169–187, 1996, J. R. Stat. Soc. B 59:475–499, 1997) and we analyze the asymptotic tail dependence of the random pair (Y
i
, Y
i + m
), in all considered cases. According to Drees (Bernoulli 9:617–657, 2003), we obtain the limit behavior of the tail empirical quantile function associated with a random sample (Y
1, Y
2,...Y
n
) and hence the asymptotic normality of a class of estimators of the tail index that includes Hill estimator.
18 citations
TL;DR: In this paper, the maximum of Y up to time T: MT = max 0 √ √ t √ T Y(t) and de termine an asymptotic expression for P (MT>u) when u → ∞.
Abstract: Define Y(t)= max 0 ≤ s ≤1 W(t+s)-W(t), where W(I‡) is a standard Wiener process. We study the maximum of Y up to time T: MT= max 0 ≤t ≤ T Y(t) and de termine an asymptotic expression for P (MT>u) when u → ∞. Further we establish the limiting Gumbel distribution of MT when T → ∞ and present the corresponding normalization sequence. © 2008 Springer Science+Business Media, LLC.
15 citations
TL;DR: The multivariate extremal index function is a direction specific extension of the well-known univariate extremals index as mentioned in this paper, which is a generalization of the extremal coefficient.
Abstract: The multivariate extremal index function is a direction specific extension of the well-known univariate extremal index. Since statistical inference on this function is difficult it is desirable to have a broad characterization of its attributes. We extend the set of common properties of the multivariate extremal index function and derive sharp bounds for the entire function given only marginal dependence. Our results correspond to certain restrictions on the two dependence functions defining the multivariate extremal index, which are opposed to Smith and Weissman’s (1996) conjecture on arbitrary dependence functions. We show further how another popular dependence measure, the extremal coefficient, is closely related to the multivariate extremal index. Thus, given the value of the former it turns out that the above bounds may be improved substantially. Conversely, we specify improved bounds for the extremal coefficient itself that capitalize on marginal dependence, thereby approximating two views of dependence that have frequently been treated separately. Our results are completed with example processes.
TL;DR: In this article, the asymptotic behavior of sample maxima of weighted Dirichlet triangular arrays was studied and the authors derived the conditions that turn such arrays in Husler-Reiss triangular arrays.
Abstract: In this paper we study the asymptotic behaviour of sample maxima of weighted Dirichlet triangular arrays. Two cases are interesting for our analysis, a) the associated random radius of the triangular array has distribution function in the Gumbel, b) or in the Weibull max-domain of attraction. In this paper we derive the asymptotic conditions that turn such arrays in Husler–Reiss triangular arrays.
TL;DR: In this article, the authors considered supercritical Markov branching processes with continuous time where every particle has one or two random scores and the class of non-degenerate limit laws for linear normed maxima is described.
Abstract: We consider supercritical Markov branching processes with continuous time where every particle has one or two random scores. We are interested in maxima of these scores over the population. The class of nondegenerate limit laws for linear normed maxima is described. Limit copulas, upper and lower tail dependence coefficients are obtained for cases of two scores and two time points. Results are illustrated by the computer simulation.
TL;DR: In this paper, the authors derived the asymptotic distributional properties of a similar estimator of a positive tail index, based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses.
Abstract: In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called “maximum likelihood” estimators of the tail index γ. In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar “maximum likelihood” estimator of a positive tail index γ, based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study.
TL;DR: In this article, a two-step estimator for the extreme value index was proposed, where the first step estimator is the Pickands estimator and the second one is the maximum likelihood estimator.
Abstract: In this paper, we build a two-step estimator $\hat{\gamma}_{\rm STEP}$
, which satisfies $\sqrt{k}(\hat{\gamma}_{\rm STEP}-\hat{\gamma}_{ML})\stackrel{P}{\rightarrow} 0$
, where $\hat{\gamma}_{ML}$
is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator $\hat{\gamma}_{\rm STEP}$
can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property.
TL;DR: In this paper, the authors follow the route through multivariate extreme value theory (EVT), which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly with Sid Resnick.
Abstract: Since the publication of his masterpiece on regular variation and its application to the weak convergence of (univariate) sample extremes in 1970, Laurens de Haan (Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam, 1970) is among the leading mathematicians in the world, with a particular focus on extreme value theory (EVT). On the occasion of his 70th birthday it is a great pleasure and a privilege to follow his route through multivariate EVT, which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly with Sid Resnick.
TL;DR: The necessary and sufficient conditions for a random variable to belong to the domains of attraction of Φα and Ψα are derived in terms of conditional moments as discussed by the authors, where Φ α is defined as a function of the conditional moment.
Abstract: Let X1, X2, ...Xn be independent and identically distributed random variables with common distribution function F. Necessary and sufficient conditions for F to belong to the domains of attraction of Φα and Ψα are derived in terms of conditional moments.
TL;DR: In this article, a detailed study of serial dependence in an ARCH(1) process from the point of view of the lower tail dependence coefficient and certain generalisations thereof is presented.
Abstract: Serial dependence in non-linear time series cannot always be reliably quantified using linear autocorrelation. We do a detailed study of serial dependence in an ARCH(1) process from the point of view of the lower tail dependence coefficient and certain generalisations thereof. Our results are relevant for estimating probabilities of consecutive value-at-risk violations in GARCH models.
TL;DR: In this paper, a CUSUM-type monitoring scheme is designed to sequentially detect changes in the regression parameter of an underlying linear model, which is used to derive the limiting extreme value distribution under the null hypothesis of structural stability.
Abstract: We study a CUSUM–type monitoring scheme designed to sequentially detect changes in the regression parameter of an underlying linear model. The test statistic used is based on recursive residuals. Main aim of this paper is to derive the limiting extreme value distribution under the null hypothesis of structural stability. The model assumptions are flexible enough to include rather general classes of error sequences such as augmented GARCH(1,1) processes. The result is underlined by an illustrative simulation study.