L.F.M. de Haan (Laurens)
http://repub.eur.nl/ppl/1896/
List of Publicationsenhttp://repub.eur.nl/logo.jpg
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. 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. 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.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.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.On bootstrap sample size in extreme value theory
http://repub.eur.nl/pub/541/
Mon, 11 Nov 2002 00:00:01 GMT<div>J.L. Geluk</div><div>L.F.M. de Haan</div>
It has been known for a long time that for bootstrapping the
probability distribution of the maximum of a sample consistently,
the bootstrap sample size needs to be of smaller order than the
original sample size. See Jun Shao and Dongsheng Tu (1995), Ex.
3.9,p. 123. We show that the same is true if we use the bootstrap
for estimating an intermediate quantile.Using a bootstrap method to choose the sample fraction in tail index estimation
http://repub.eur.nl/pub/12389/
Thu, 01 Feb 2001 00:00:01 GMT<div>J. Danielsson</div><div>L. Peng</div><div>C.G. de Vries</div><div>L.F.M. de Haan</div>
Tail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e., the number of extreme order statistics on which the estimation is based. A complete solution to the sample fraction selection is given by means of a two-step subsample bootstrap method. This method adaptively determines the sample fraction that minimizes the asymptotic mean-squared error. Unlike previous methods, prior knowledge of the second-order parameter is not required. In addition, we are able to dispense with the need for a prior estimate of the tail index which already converges roughly at the optimal rate. The only arbitrary choice of parameters is the number of Monte Carlo replications.A bootstrap-based method to achieve optimality on estimating the extreme-value index
http://repub.eur.nl/pub/1650/
Thu, 25 May 2000 00:00:01 GMT<div>G. Draisma</div><div>L.F.M. de Haan</div><div>L. Peng</div><div>T.T. Pereira</div>
Estimators of the extreme-value index are based on a set of upper order statistics. We present an adaptive method to choose the number of order statistics involved in an optimal way, balancing variance and bias components. Recently this has been achieved for the similar but somewhat less involved case of regularly varying tails (Drees and Kaufmann(1997); Danielsson et al.(1996)). The present paper follows the line of proof of the last mentioned paper.Using a bootstrap method to choose the sample fraction in tail index estimation
http://repub.eur.nl/pub/1652/
Thu, 25 May 2000 00:00:01 GMT<div>J. Danielsson</div><div>L.F.M. de Haan</div><div>L. Peng</div><div>C.G. de Vries</div>
Tail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e. the number of extreme order statistics on which the estimation is based. A complete solution to the sample fraction selection is given by means of a two step subsample bootstrap method. This method adaptively determines the sample fraction that minimizes the asymptotic mean squared error. Unlike previous methods, prior knowledge of the second order parameter is not required. In addition, we are able to dispense with the need for a prior estimate of the tail index which already converges roughly at the optimal rate. The only arbitrary choice of parameters is the number of Monte Carlo replications.Penultimate Approximation for Hill's Estimator
http://repub.eur.nl/pub/7716/
Wed, 11 Aug 1999 00:00:01 GMT<div>S. Cheng</div><div>L.F.M. de Haan</div>
We prove that the probability distribution of Hill's estimator can be better approximated by a series of appropriate gamma distributions than by the limiting normal distribution.How to make a Hill Plot
http://repub.eur.nl/pub/7748/
Thu, 27 Aug 1998 00:00:01 GMT<div>H. Drees</div><div>L.F.M. de Haan</div><div>S. Resnick</div>
An abundance of high quality data sets requiring heavy tailed models necessitates reliable methods of estimating the shape parameter governing the degree of tail heaviness. The Hill estimator is a popular method for doing this but its practical use is encumbered by several difficulties. We show that an alternative method of plotting Hill estimator values is more revealing than the standard method unless the underlying data comes from a Pareto distribution.Approximation by Penultimate Extreme Value Distributions
http://repub.eur.nl/pub/7760/
Mon, 23 Mar 1998 00:00:01 GMT<div>L.F.M. de Haan</div>
n certain cases the distribution of the normalized maximum of a sample can be better approximated by a sequence of different extreme value distributions than by the final one. We show that these cases are rather restricted and that the possible improvement is not spectacular.Approximation by Penultimate Stable Laws
http://repub.eur.nl/pub/7793/
Thu, 04 Sep 1997 00:00:01 GMT<div>L.F.M. de Haan</div><div>L. Peng</div><div>H. Iglesias Pereira</div>
In certain cases partial sums of i.i.d. random variables with finite variance are better approximated by a sequence of stable distributions with indices \\alpha_n \\to 2 than by a normal distribution. We discuss when this happens and how much the convergence rate can be improved by using penultimate approximations. Similar results are valid for other stable distributions.Stable Probability Distributions and their Domains of Attraction
http://repub.eur.nl/pub/7796/
Fri, 15 Aug 1997 00:00:01 GMT<div>J.L. Geluk</div><div>L.F.M. de Haan</div>
The theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition.Using a Bootstrap Method to choose the Sample Fraction in Tail Index Estimation
http://repub.eur.nl/pub/7806/
Wed, 29 Jan 1997 00:00:01 GMT<div>J. Danielsson</div><div>L.F.M. de Haan</div><div>L. Peng</div><div>C.G. de Vries</div>
We use a subsample bootstrap method to get a consistent estimate of the asymptotically optimal choice of the sample fraction, in the sense of minimal mean squared error, which is needed for tail index estimation. Unlike previous methods our procedure is fully self contained. In particular, the method is not conditional on an initial consistent estimate of the tail index; and the ratio of the first and second order tail indices is left unrestricted, but we require the ratio to be strictly positive. Hence the current method yields a complete solution to tail index estimation as it is not predicated on a more or less arbitrary choice of the number of highest order statistics.Extremal behavior of solutions to a stochastic difference equation, with applications to ARCH processes
http://repub.eur.nl/pub/12438/
Sun, 01 Jan 1989 00:00:01 GMT<div>L.F.M. de Haan</div><div>S. Resnick</div><div>H. Rootzen</div><div>C.G. de Vries</div>