L.F.M. de Haan (Laurens)
http://repub.eur.nl/ppl/1896/
List of Publicationsenhttp://repub.eur.nl/eur_logo.png
http://repub.eur.nl/
RePub, Erasmus University RepositoryThe number of active bidders in internet auctions
http://repub.eur.nl/pub/40955/
Mon, 01 Jul 2013 00:00:01 GMT<div>L.F.M. de Haan</div><div>C.G. de Vries</div><div>C. Zhou</div>
Internet auctions attract numerous agents, but only a few become active bidders. Under the Independent Private Values Paradigm the valuations of the active bidders form a specific record sequence. This record sequence implies that if the number n of potential bidders is large, the number of active bidders is approximately 2log. n, potentially explaining the relative inactivity. Moreover, if the arrival of potential bidders forms a non-homogeneous Poisson process due to a time preference for auctions that are soon to close, then the arrival process of the active bidders is approximately a Poisson arrival process. Estimating extreme bivariate quantile regions
http://repub.eur.nl/pub/40557/
Sat, 01 Jun 2013 00:00:01 GMT<div>J.H.J. Einmahl</div><div>L.F.M. de Haan</div><div>A. Krajina</div>
When simultaneously monitoring two possibly dependent, positive risks one is often interested in quantile regions with very small probability p. These extreme quantile regions contain hardly any or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we need to deal with probabilities p < 1/n, with n the sample size. We consider quantile regions of the form {(x, y) ∈ (0, ∞)2: f(x, y) ≤ β}, where f, the joint density, is decreasing in both coordinates. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. A detailed simulation study shows the very good statistical performance of the estimated quantile regions. We also apply the method to find extreme risk regions for bivariate insurance claims. Bias correction in extreme value statistics with index around zero
http://repub.eur.nl/pub/38681/
Thu, 20 Sep 2012 00:00:01 GMT<div>J.J. Cai</div><div>L.F.M. de Haan</div><div>C. Zhou</div>
Applying extreme value statistics in meteorology and environmental science requires accurate estimators on extreme value indices that can be around zero. Without having prior knowledge on the sign of the extreme value indices, the probability weighted moment (PWM) estimator is a favorable candidate. As most other estimators on the extreme value index, the PWM estimator bears an asymptotic bias. In this paper, we develop a bias correction procedure for the PWM estimator. Moreover, we provide bias-corrected PWM estimators for high quantiles and, when the extreme value index is negative, the endpoint of a distribution. The choice of k, the number of high order statistics used for estimation, is crucial in applications. The asymptotically unbiased PWM estimators allows the choice of higher level k, which results in a lower asymptotic variance. Moreover, since the bias-corrected PWM estimators can be applied for a wider range of k compared to the original PWM estimator, one gets more flexibility in choosing k for finite sample applications. All advantages become apparent in simulations and an environmental application on estimating "once per 10,000 years" still water level at Hoek van Holland, The Netherlands. Exceedance probability of the integral of a stochastic process
http://repub.eur.nl/pub/68076/
Wed, 01 Feb 2012 00:00:01 GMT<div>A. Ferreira</div><div>L.F.M. de Haan</div><div>C. Zhou</div>
Let X={X(s)}s∈S be an almost sure continuous stochastic process (S compact subset of Rd) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of ∫SX(s)ds, which turns out to be of Generalized Pareto type with an extra 'spatial' parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall. The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.Extreme residual dependence for random vectors and processes
http://repub.eur.nl/pub/25626/
Tue, 01 Mar 2011 00:00:01 GMT<div>L.F.M. de Haan</div><div>C. Zhou</div>
A two-dimensional random vector in the domain of attraction of an extreme value distribution G is said to be asymptotically independent (i.e. in the tail) if G is the product of its marginal distribution functions. Ledford and Tawn (1996) discussed a form of residual dependence in this case. In this paper we give a characterization of this phenomenon (see also Ramos and Ledford (2009)), and offer extensions to higher-dimensional spaces and stochastic processes. Systemic risk in the banking system is treated in a similar framework. Stationary max-stable fields associated to negative definite functions
http://repub.eur.nl/pub/71478/
Tue, 01 Sep 2009 00:00:01 GMT<div>Z. Kabluchko</div><div>M. Schlather</div><div>L.F.M. de Haan</div>
Let Wi, i ∈ N{double struck}, be independent copies of a zero-mean Gaussian process {W(t), t ∈ R{double struck}d} with stationary increments and variance σ2(t). Independently of Wi, let ∑∞ i=1 δUi be a Poisson point process on the real line with intensity e-y dy. We show that the law of the random family of functions {Vi(·), i ∈ N{double struck}}, where Vi(t) = Ui + Wi(t) - σ2(t)/2, is translation invariant. In particular, the process η(t) = V∞ i=1 Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n →∞if and only if W is a (nonisotropic) fractional Brownian motion on R{double struck}d. Under suitable conditions on W, the process η has a mixed moving maxima representation.A test procedure for detecting super-heavy tails
http://repub.eur.nl/pub/14326/
Sun, 01 Feb 2009 00:00:01 GMT<div>I. Alves</div><div>L.F.M. de Haan</div><div>C. Neves</div>
The aim of this work is to develop a test to distinguish between heavy and super-heavy tailed probability distributions. These classes of distributions are relevant in areas such as telecommunications and insurance risk, among others. By heavy tailed distributions we mean probability distribution functions with polynomially decreasing upper tails (regularly varying tails). The term super-heavy is reserved for right tails decreasing to zero at a slower rate, such as logarithmic, or worse (slowly varying tails). Simulations are presented for several models and an application with telecommunications data is provided.Parametric tail copula estimation and model testing
http://repub.eur.nl/pub/64185/
Tue, 01 Jul 2008 00:00:01 GMT<div>L.F.M. de Haan</div><div>C. Neves</div><div>L. Peng</div>
Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test.The Extent of Internet Auction Markets
http://repub.eur.nl/pub/13676/
Thu, 17 Apr 2008 00:00:01 GMT<div>L.F.M. de Haan</div><div>C.G. de Vries</div><div>C. Zhou</div>
Internet auctions attract numerous agents, but only a few become active bidders. A major difficulty in the structural analysis of internet auctions is that the number of potential bidders is unknown. Under the independent private value paradigm (IPVP)the valuations of the active bidders form a specific record sequence. This record sequence implies that if the number n of potential bidders is large, the number of active bidders is approximately 2 log n, explaining the relative inactivity. Empirical evidence for the 2 log n rule is provided. This evidence can also be interpreted as a weak test of the IPVP.The expected payoff to Internet auctions
http://repub.eur.nl/pub/14192/
Tue, 01 Jan 2008 00:00:01 GMT<div>L.F.M. de Haan</div><div>C.G. de Vries</div><div>C. Zhou</div>
In an Internet auction, the expected payoff acts as a benchmark of the reasonableness of the price that is paid for the purchased item. Since the number of potential bidders is not observable, the expected payoff is difficult to estimate accurately. We approach this problem by considering the bids as a record and 2-record sequence of the potential bidder's valuation and using the Extreme Value Theory models to model the tail distribution of the bidder's valuation and study the expected payoff. Along the discussions for three different cases regarding the extreme value index γ, we show that the observed payoff does not act as an accurate estimation of the expected payoff in all the cases except a subclass of the case γ = 0. Within this subclass and under a second order condition, the observed payoff consistently converges to the expected payoff and the corresponding asymptotic normality holds.Mixed moment estimator and location invariant alternatives
http://repub.eur.nl/pub/14547/
Tue, 01 Jan 2008 00:00:01 GMT<div>M.I. Fraga Alves</div><div>M.I. Gomes</div><div>L.F.M. de Haan</div><div>C. Neves</div>
A new class of estimators of the extreme value index is developed. It has a simple form and is asymptotically very close to the maximum likelihood estimator for a wide class of heavy-tailed models. We also propose an alternative class of estimators, dependent on a tuning parameter p ∈ (0,1) and invariant for changes in both scale and/or location. Such a tuning parameter can help us to choose the number of top order statistics to be used in the estimation of extreme parameters.Comments on "Plotting positions in extreme value analysis"
http://repub.eur.nl/pub/53579/
Thu, 01 Mar 2007 00:00:01 GMT<div>L.F.M. de Haan</div>
Weak & Strong Financial Fragility
http://repub.eur.nl/pub/8747/
Wed, 14 Feb 2007 00:00:01 GMT<div>J.L. Geluk</div><div>L.F.M. de Haan</div><div>C.G. de Vries</div>
The stability of the financial system at higher loss levels is either characterized by asymptotic dependence or asymptotic independence. If asymptotically independent, the dependency, when present, eventually dies out completely at the more extreme quantiles, as in case of the multivariate normal distribution. Given that financial service firms' equity returns depend linearly on the risk drivers, we show that the marginals' distributions maximum domain of attraction determines the type of systemic (in-)stability. A scale for the amount of dependency at high loss lovels is designed. This permits a characterization of systemic risk inherent to different financial network structures. The theory also suggests the functional form of the economically relevant limit copulas.Approximations to the tail empirical distribution function with application to testing extreme value conditions
http://repub.eur.nl/pub/69283/
Sun, 01 Oct 2006 00:00:01 GMT<div>H. Drees</div><div>L.F.M. de Haan</div><div>D. Li</div>
Weighted approximations to the tail of the distribution function and its empirical counterpart are derived which are suitable for applications in extreme value statistics. The approximation of the tail empirical distribution function is then used to develop an Anderson-Darling type test of the null hypothesis that the distribution function belongs to the domain of attraction of an extreme value distribution.A class of distribution functions with less bias in extreme value estimation
http://repub.eur.nl/pub/69281/
Fri, 01 Sep 2006 00:00:01 GMT<div>L.F.M. de Haan</div><div>L. Canto e Castro</div>
Let X1, X2, ... be i.i.d. random variables and let their distribution be in the domain of attraction of an extreme value distribution. Quite a few estimators of the extreme value index are known to be consistent under the domain of attraction conditions. When it comes to asymptotic normality a condition that is called second-order condition is very useful. The condition yields a speed of convergence of a polynomial rate. Then one gets asymptotically a normal distribution without bias, provided one restricts the number of tail observations used in the estimation to a certain polynomial of n, the total number of observations. We investigate what happens if the speed of convergence is faster than any polynomial rate. In that case one can use many more tail observations without creating bias.Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition
http://repub.eur.nl/pub/65822/
Tue, 01 Aug 2006 00:00:01 GMT<div>J.H.J. Einmahl</div><div>L.F.M. de Haan</div><div>D. Li</div>
Consider n i.i.d. random vectors on ℝ 2, with unknown, common distribution function F. Under a sharpening of the extreme value condition on F, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution. We derive the limiting distribution of the test statistic from the aforementioned weighted approximation. This limiting distribution contains unknown functional parameters. Therefore, we show that a version with estimated parameters converges weakly to the true limiting distribution. Based on this result, the finite sample properties of our testing procedure are investigated through a simulation study. A real data application is also presented.Discussion of "copulas: Tales and facts", by Thomas Mikosch
http://repub.eur.nl/pub/53824/
Wed, 01 Mar 2006 00:00:01 GMT<div>L.F.M. de Haan</div>
Spatial extremes: Models for the stationary case
http://repub.eur.nl/pub/73283/
Wed, 01 Feb 2006 00:00:01 GMT<div>L.F.M. de Haan</div><div>T. Pereira</div>
The aim of this paper is to provide models for spatial extremes in the case of stationarity. The spatial dependence at extreme levels of a stationary process is modeled using an extension of the theory of max-stable processes of de Haan and Pickands [Probab. Theory Related Fields 72 (1986) 477-492]. We propose three one-dimensional and three two-dimensional models. These models depend on just one parameter or a few parameters that measure the strength of tail dependence as a function of the distance between locations. We also propose two estimators for this parameter and prove consistency under domain of attraction conditions and asymptotic normality under appropriate extra conditions.On maximum likelihood estimation of the extreme value index
http://repub.eur.nl/pub/74963/
Sun, 01 Aug 2004 00:00:01 GMT<div>H. Drees</div><div>A. Ferreira</div><div>L.F.M. de Haan</div>
We prove asymptotic normality of the so-called maximum likelihood estimator of the extreme value index.Bivariate tail estimation: Dependence in asymptotic independence
http://repub.eur.nl/pub/65618/
Thu, 01 Apr 2004 00:00:01 GMT<div>G. Draisma</div><div>H. Drees</div><div>A. Ferreira</div><div>L.F.M. de Haan</div>
In the classical setting of bivariate extreme value theory, the procedures for estimating the probability of an extreme event are not applicable if the componentwise maxima of the observations are asymptotically independent. To cope with this problem, Ledford and Tawn proposed a submodel in which the penultimate dependence is characterized by an additional parameter. We discuss the asymptotic properties of two estimators for this parameter in an extended model. Moreover, we develop an estimator for the probability of an extreme event that works in the case of asymptotic independence as well as in the case of asymptotic dependence, and prove its consistency.