Likelihood estimation of the extremal index

The article develops the approach of Ferro and Segers (J.R. Stat. Soc., Ser. B 65:545, 2003) to the estimation of the extremal index, and proposes the use of a new variable decreasing the bias of the likelihood based on the point process character of the exceedances. Two estimators are discussed: a maximum likelihood estimator and an iterative least squares estimator based on the normalized gaps between clusters. The first provides a flexible tool for use with smoothing methods. A diagnostic is given for condition , under which maximum likelihood is valid. The performance of the new estimators were tested by extensive simulations. An application to the Central England temperature series demonstrates the use of the maximum likelihood estimator together with smoothing methods.      

Capturing the multivariate extremal index: bounds and interconnections

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.      

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.      

On approximating max-stable processes and constructing extremal copula functions

There is an infinite number of parameters in the definition of multivariate maxima of moving maxima (M4) processes, which poses challenges in statistical applications where workable models are preferred. This paper establishes sufficient conditions under which an M4 process with infinite number of parameters may be approximated by an M4 process with finite number of parameters. In statistical inferences, the paper focuses on a family of sectional multivariate extreme value copula (SMEVC) functions which is derived from the joint distribution functions of M4 processes. A new non-standard parameter estimation procedure is introduced, which is based on order statistics of ratios of (transformed) marginal unit Fréchet random variables, and is shown via simulation to be more efficient than a semi-parametric estimation procedure. In real data analysis, empirical results show that SMEVCs are more flexible for modeling various dependence structures, and perform better than the widely used Gumbel-Hougaard copulas.     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

Boundedness of extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^∗$u^{\ast }$ in dimension N=4$N=4$ of the semilinear elliptic equation −Δu=λf(u)$-\Delta u=\lambda f\left(u\right)$, in a general smooth bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, with Dirichlet data u_|∂Ω=0$u{|}_{\partial \Omega }=0$, where f$f$ is a C¹$C^1$ positive, nondecreasing and convex function in [0,∞)$[0,\infty )$ such that f(s)/s→∞$f\left(s\right)/s\to \infty$ as s→∞$s\to \infty$.     

Asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three

Asymptotic properties of extremal solutions of linear inclusions of order three with zero Lyapunov exponent are investigated. Under certain conditions it is shown that all extremal solutions of such inclusions tend to the same (up to a multiplicative factor) solution, which is central symmetric. The structure of the convex set of extremal norm is studied. A number of extremal points of this set are described.     

An extremal problem for harmonic self-mappings of the unit disk

Let AutH(D) be the class of harmonic automorphisms of the unit disk D. For the upper bounds of are given.     

Regularly varying random fields

We study the extremes of multivariate regularly varying random fields. The crucial tools in our study are the tail field and the spectral field, notions that extend the tail and spectral processes of Basrak and Segers (2009). The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown–Resnick random fields.     

Problems and results in extremal combinatorics—II

Extremal Combinatorics is one of the central areas in Discrete Mathematics. It deals with problems that are often motivated by questions arising in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics considered here include questions in Extremal Graph Theory, Polyhedral Combinatorics and Probabilistic Combinatorics. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably biased, and as the title of the paper suggests, it is a sequel to a previous paper [N. Alon, Problems and results in extremal combinatorics—I, Discrete Math. 273 (2003), 31-53.] of the same flavor, and hopefully a predecessor of another related future paper. Each section of this paper is essentially self contained, and can be read separately.     

On Mono- and Epimorphisms in Varieties of Ordered Algebras

We study morphisms in varieties of ordered universal algebras. We prove that (i) monomorphisms are precisely the injective homomorphisms and that (ii) every regular monomorphism is an order embedding, but the converse is not true in general. We also give a necessary and sufficient condition for a morphism to be a regular epimorphism. Finally, we discuss factorizations in such varieties.     

Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities

We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin.

     

Contravariant forms and extremal projectors

Tensor product of irreducible modules of highest weight over a semi-simple quantum group is completely reducible if and only if a natural contravariant form is non-degenerate when restricted to the span of singular vectors. We express this restriction through the extremal projector of the quantum group providing a computationally feasible criterion for complete reducibility of tensor products.     

Tail and dependence behavior of levels that persist for a fixed period of time

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 Fréchet 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. Research partially supported by FCT/POCTI and POCI/FEDER.     

On the existence and nonexistence of extremal metrics on toric Kähler surfaces

In this paper we study the existence of extremal metrics on toric Kähler surfaces. We show that on every toric Kähler surface, there exists a Kähler class in which the surface admits an extremal metric of Calabi. We found a toric Kähler surface of 9 -fixed points which admits an unstable Kähler class and there is no extremal metric of Calabi in it. Moreover, we prove a characterization of the K-stability of toric surfaces by simple piecewise linear functions. As an application, we show that among all toric Kähler surfaces with 5 or 6 -fixed points, is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature.     

Lower tail dependence for Archimedean copulas: Characterizations and pitfalls

Tail dependence copulas provide a natural perspective from which one can study the dependence in the tail of a multivariate distribution. For Archimedean copulas with continuously differentiable generators, regular variation of the generator near the origin is known to be closely connected to convergence of the lower tail dependence copulas to the Clayton copula. In this paper, these characterizations are refined and extended to the case of generators which are not necessarily continuously differentiable. Moreover, a counterexample is constructed showing that even if the generator of a strict Archimedean copula is continuously differentiable and slowly varying at the origin, then the lower tail dependence copulas still do not need to converge to the independent copula.     

On the girth of extremal graphs without shortest cycles

Let EX(ν;{C₃,…,C_n}) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n+1 cycle? We prove that the diameter of G is at most n−1, and present several results concerning the above question: the girth of G is g=n+1 if (i) ν≥n+5, diameter equal to n−1 and minimum degree at least 3; (ii) ν≥12, ν∉{15,80,170} and n=6. Moreover, if ν=15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if ν≥2n−3 and n≥7 the girth is at most 2n−5. We also show that the answer to the question is negative for ν≤n+1+⌊(n−2)/2⌋.     

Construction of extremal Type II Z2k-codes

We give methods for constructing many self-dual ${\mathbb{Z}}_m$-codes and Type II ${\mathbb{Z}}_{2k}$-codes of length 2n starting from a given self-dual ${\mathbb{Z}}_m$-code and Type II ${\mathbb{Z}}_{2k}$-code of length 2n, respectively. As an application, we construct extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 24 for $k=4,5,\dots ,20$ and extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 32 for $k=4,5,\dots ,10$. We also construct new extremal Type II ${\mathbb{Z}}_4$-codes of lengths 56 and 64.