An interview with Laurens de Haan

Laurens de Haan was born January 15, 1937 in Rotterdam, The Netherlands. He graduated 1966 in mathematics and received a doctoral degree in 1970 from the University of Amsterdam, while working at the Mathematical center CWI in Amsterdam. Since 1973 he was Professor for probability and mathematical statistics at the Econometric Institute of the Economic Faculty at the Erasmus University Rotterdam, where he retired 1998. Since 2008 he is part-time professor at the Department of Econometrics and Operations Research of Tilburg University. Laurens de Haan has been active in research throughout his career. He has published more than 110 scientific papers. Among other distinctions, he was elected IMS fellow for his seminal contributions to extreme value theory in 1977, and he was appointed Honorary Doctor of the University of Lisbon in 2000.     

Statistics of extremes for IID data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions

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. To Laurens de Haan, a token of friendship.     

It was 30 years ago today when Laurens de Haan went the multivariate way

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.      

Bootstrap and empirical likelihood methods in extremes

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. Dedicated to Professor Laurens de Haan on the occasion of his 70th birthday.     

Reduced-Load Equivalence for Queues with Gaussian Input

In this note, we consider a queue fed by a number of independent heterogeneous Gaussian sources. We study under what conditions a reduced load equivalence holds, i.e., when a subset of the sources becomes asymptotically dominant as the buffer size increases. For this, recent results on extremes of Gaussian processes [6] are combined with de Haan theory. We explain how the results of this note relate to square root insensitivity and moderately heavy tails.The research was supported by the Netherlands Organization for Scientific Research (NWO) under grant 631.000.002.This revised version was published online in June 2005 with corrected coverdate     

On testing extreme value conditions

Applications of univariate extreme value theory rely on certain as- sumptions. Recently, two methods for testing these extreme value conditions are derived by [Dietrich, D., de Haan, L., Hüsler, J., Extremes 5: 71–85, (2002)] and [Drees, H., de Haan, L., Li, D., J. Stat. Plan. Inference, 136: 3498–3538, (2006)]. In this paper we compare the two tests by simulations and investigate the effect of a possible weight function by choosing a parameter, the test error and the power of each test. The conclusions are useful for extreme value applications.     

Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.     

A two-step estimator of the extreme value index

In this paper, we build a two-step estimator , which satisfies , where is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator 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. The author thanks Laurens de Haan for motivating this work and giving helpful comments. The author also thanks two anonymous referees for their useful comments.     

Some aspects of extreme value statistics under serial dependence

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.      

Limit theorems of general functions of independent and identically distributed random variables

In this paper we derive limit theorems of some general functions of independent and identically distributed random variables. A stability property is used to derive the limit theory for general functions. A procedure followed in de Haan (1976) and Leadbetter et al. (1983) is used to prove the main result. The limit theorems for the maximum, minimum and sum of fixed sample sizes and random sample sizes are derived as special cases of the main result.     

Some properties of entire functions with nonnegative Taylor coefficients

If f is an entire function of arbitrary finite order and with nonnegative Taylor coefficients, then we prove that its restriction to the positive part of the real axis belongs to de Haan’s class Γ. We also show that f/f′ is a Beurling slowly varying function.     

Quasicrystals—The impact of N.G. de Bruijn

In this paper we put the work of Professor N.G. de Bruijn on quasicrystals in historical context. After briefly discussing what went before, we shall review de Bruijn’s work together with recent related theoretical and experimental developments. We conclude with a discussion of Yang–Baxter integrable models on Penrose tilings, for which essential use of de Bruijn’s work has been made.     

Prevalence of hyperbolicity for complex singular cycles

In this article we consider two kinds of complex singular cycles arising for vector fields defined on three-dimensional manifolds. We prove that, under some generic conditions, any one parameter family of vector fields passing through these cycles has the following property: Hyperbolicity is a prevalent phenomena.Dedicated to the memory of Professor R. Mañé.Partially Supported by Fondecyt 1941080 and Dirección de Investigación Universidad de Santiago de Chile     

Local asymptotic normality in a stationary model for spatial extremes

De Haan and Pereira (2006) [6] provided models for spatial extremes in the case of stationarity, which depend on just one parameter β>0 measuring tail dependence, and they proposed different estimators for this parameter. We supplement this framework by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of β. It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance, if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.     

Characterizing the inf-sup condition on product spaces

In this paper we establish characterization results for the continuous and discrete inf-sup conditions on product spaces. The inf-sup condition for each component of the bilinear form involved and suitable decompositions of the pivot space in terms of the associated null spaces are the key ingredients of our theorems. We illustrate the theory through its application to bilinear forms arising from the variational formulations of several boundary value problems. Dedicated to Professor Ivo Babuska on the occasion of his 82nd birthday. This research was partially supported by Centro de Modelamiento Matemático (CMM) of the Universidad de Chile, by Centro de Investigación en Ingenierí a Matemática (CI²MA) of the Universidad de Concepción, by FEDER/MCYT Project MTM2007-63204, and by Gobierno de Aragón (Grupo Consolidado PDIE).     

Unitary colligations of operators in Krein spaces

A model for non-contractive functions is given, according to which every bounded analytic function coincides with the characteristic function of a suitable unitary colligation. In our construction, the function is expressed in terms of a fundamentally reducible unitary dilation of the basic operator of the colligation.The results in this paper first appeared as part of the author's doctoral thesis [13] at the Universidad Central de Venezuela, under the supervision of Mischa Cotlar. The work has been supported by the Consejo Nacional de Investigaciones Cientificas y Techológicas (CONICIT-Venezuela). I am grateful to Professor Aad Dijksma, who read an earlier version of this note and made several helpful criticisms. I also thank Professor Mischa Cotlar for his kind encouragement and Professor Alejandra Cabana for correcting the english text.     

Copula convergence theorems for tail events

Tail dependence is studied from a distributional point of view by means of appropriate copulae. We derive similar results to the famous Pickands–Balkema–de Haan Theorem of Extreme Value Theory. Under regularity conditions, it is shown that the Clayton copula plays among the family of archimedean copulae the role of the generalized Pareto distribution. The practical usefulness of the results is illustrated in the analysis of stock market data.     

On the factorization of Abelian groups

In this note we consider a conjecture made by Professor de Bruijn and subsequent generalizations of this conjecture. We show by using constructions of his that in a certain sense these generalizations are the best possible. A survey of these known results is presented. Finally some new results of this kind are given.     

A characterization of the one parameter exponential family of distributions by monotonicity of likelihood ratios

Summary Any one parameter exponential family of distributions has monotone likelihood ratios. As the product probabilities of n identical distributions of an exponential family form again an exponential family, it has monotone likelihood ratios for arbitrary n. Furthermore, the members of an exponential family are mutually absolutely continuous. In Part 1, we show that these properties uniquely characterize the exponential family. The application of this result to the theory of testing hypotheses (Part 2) shows that if a family of mutually absolutely continuous distributions has uniformly most powerful tests for arbitrary levels of significance, and arbitrary sample sizes, then it is necessarily an exponential family.The research was done while this author was a Visiting Professor in the Department of Statistics at the University of Chicago. It was supported by Research Grants Nos. NSF-G10368 and NSF-G21058 from the Division of Mathematical, Physical and Engineering Sciences of the National Science Foundation.     

Navier-Stokes方程稳定性研究(Ⅱ)

本文对Navier-Stokes方程与热传导方程的性质进行了比较。法国数学家、偏微分方程权威J.Leray教授在其对Navier-Stokes方程的研究中,曾由热传导方程出发而求得Navier-Stokes方程某种初(边)值问题的适定性结果[2].巴黎十一大学的R.Temam等专家、教授也曾多次提出过将两类方程类比的疑问。本文试将其中根本不同点做了叙述和例证。