Review of testing issues in extremes: in honor of Professor Laurens de Haan

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.      

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.     

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.     

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.     

The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on extreme value theory (EVT) has found a successful domain of application in such a context, outperforming other methods. Given a parametric model provided by EVT, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the maximum Lq-likelihood estimator (MLqE), introduced by Ferrari and Yang (Estimation of tail probability via the maximum Lq-likelihood method, Technical Report 659, School of Statistics, University of Minnesota, 2007 ). We show that the MLqE outperforms the standard MLE, when estimating tail probabilities and quantiles of the generalized extreme value (GEV) and the generalized Pareto (GP) distributions. First, we assess the relative efficiency between the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q→1, the new estimator approaches the traditional maximum likelihood estimator (MLE), recovering its desirable asymptotic properties; when q ≠ 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (mean squared error).      

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.      

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.     

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.     

Smoothing the Moment Estimator of the Extreme Value Parameter

Let {X _n be a sequence of i.i.d. random variables whose common distribution F belongs to the domain of attraction of an extreme value law. A semi-parametric estimator of the extreme value parameter is the Dekkers, Einmahl and de Haan [8] moment estimator. Practical use of this estimator requires the problematic choice of a number k=k(n) of upper order statistics and there are few reliable guidelines for this choice. An averaging or smoothing technique is proposed for this estimator yielding a less volatile function of k which in practice aids estimation.     

Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity

The paper gives sufficient conditions for domains of attraction of multivariate extreme value distributions. Under the assumption of absolute continuity of a multivariate distribution, the criteria enable one to examine, by using limits of some rescaled conditional densities, whether the distribution belongs to the domain of attraction of some multivariate extreme value distribution. If this is the case, the criteria also determine how to construct such an extreme value distribution. Unlike the criterion given by de Haan and Resnick [1987,Stochastic Process. Appl.2583–93], the criteria are easily applicable even when the marginal tails are not Pareto-like.     

On approximation of meromorphic functions having index-pair (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

The purpose of this paper is to generalize the results obtained by Winiarski (Ann. Polon. Math. 29:259–273, 1970) and Kasana and Kumar (Publ. Mat. 38:255–267, 1994) for the M ₀(C) of all entire functions onto the class M _m (C), m ≥ 0 of all meromorphic functions with exactly m poles on the complex plane C.     

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form

Distributive Lattices with a Generalized Implication: Topological Duality

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.     

Deducing the multidimensional Szemerédi theorem from an infinitary removal lemma

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T ₁, T ₂, …, T _d : ℤ ↷ (X, ∑, μ) ([6]), and so, via the Furstenberg correspondence principle introduced in [5], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserving system that underlies the Furstenberg-Katznelson analysis, instead modifying an approach recently developed in [1] to pass to a large extension of our original system in which this analysis greatly simplifies. The proof is then completed using an adaptation of arguments developed by Tao in [13] for his study of an infinitary analog of the hypergraph removal lemma. In a sense, this addresses the difficulty, highlighted by Tao, of establishing a direct connection between his infinitary, probabilistic approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic approach to Szemerédi’s Theorem set in motion by Furstenberg [5].     

Topological cones: functional analysis in a T<Subscript>0</Subscript>-setting

Already in his PhD Thesis on compact Abelian semigroups under the direction of Karl Heinrich Hofmann the author was lead to investigate locally compact cones (Keimel in Math. Z. 99:205–428, 1967). This happened in the setting of Hausdorff topologies. The theme of topological cones has been reappearing in the author’s work in a non-Hausdorff setting motivated by the needs of mathematical models for a denotational semantics of languages combining probabilistic and nondeterministic choice. This is in the line of common work with Karl Heinrich Hofmann in Continuous Lattices and Domains (Gierz et al. in Encyclopedia of Mathematics and its Applications, vol. 93, 2003). Domain Theory is based on order theoretical notions from which intrinsic non-Hausdorff topologies are derived. Along these lines, domain theoretical variants of (sub-) probability measures have been introduced by Jones and Plotkin (Jones, PhD thesis, 1990; Jones and Plotkin in Proceedings of the Fourth Annual Symposium on Logic in Computer Science, pp. 186–195, 1989). Kirch (Master’s thesis, 1993) and Tix (Master’s thesis, 1995) have extended this theory to a domain theoretical version of measures and they have introduced and studied directed complete partially ordered cones as appropriate structures. Driven by the needs of a semantics for languages combining probability and nondeterminism, Tix (Theor. Comput. Sci 264:205–218, 1999; PhD thesis, 1999) and later on Plotkin and Keimel (Electron. Notes Theor. Comput. Sci. 129:1–104, 2005) developed basic functional analytic tools for these structures. In this paper we extend this theory to topological cones the topologies of which are strongly non-Hausdorff. We carefully introduce these structures and their elementary properties. We prove Hahn-Banach type separation theorems under appropriate local convexity hypotheses. We finally construct a monad assigning to every topological cone C another topological cone the elements of which are nonempty compact convex subsets of C. For proving that this construction has good properties needed for the application in semantics we use the functional analytic tools developed before. Dedicated to Karl Heinrich Hofmann at the occasion of his 75th birthday. Thanks to Gordon Plotkin for numerous discussions. Preliminary results have been announced at MFPS XXIII 20. In his Master’s thesis supervised by the author, B. Cohen 6 has worked out some of those results.     

On Quasi-invariance of Infinite Product Measures

Quasi-invariance of infinite product measures is studied when a locally compact second countable group acts on a standard Borel space. A characterization of l ²-quasi-invariant infinite product measures is given. The group that leaves the measure class invariant is also studied. In the case where the group acts on itself by translations, our result extends previous ones obtained by Shepp (Ann. Math. Stat. 36:1107–1112, 1965) and by Hora (Math. Z. 206:169–192, 1991; J. Theor. Probab. 5:71–100, 1992) to all connected Lie groups.      

Limit theorems for records from discrete distributions

Weak and strong functional limit theorems are obtained for record values and record epochs in a sequence of independent random variables with common distribution F. The emphasis is on the case in which F is concentrated on the non-negative integers. For contrast, the well-known case of continuous F is also considered. Analogues of results obtained earlier by Resnick, de Haan and the author for continuous F are presented here for F concentrated on the non-negative integers. Also is investigated under which circumstances the latter case is so close to the continuous F case that the resulting limit theorems are the same.     

Weak Type Inequality for a Family of Singular Integral Operators Related with the Gaussian Measure

In this paper we study a family of singular integral operators that generalizes the higher order Gaussian Riesz Transforms and find the right weight w to make them continuous from L ¹(wdγ) into L ^1, ∞(dγ), being Some boundedness properties of these operators had already been derived by Urbina (Ann Scuola Norm Sup Pisa Cl Sci 17(4):531–567, 1990) and Pérez (J Geom Anal 11(3):491–507, 2001).