Bayesian inference for clustered extremes

We consider Bayesian inference for the extremes of dependent stationary series. We discuss the virtues of the Bayesian approach to inference for the extremal index, and for related characteristics of clustering behaviour. We develop an inference procedure based on an automatic declustering scheme, and using simulated data we implement and assess this procedure, making inferences for the extremal index, and for two cluster functionals. We then apply our procedure to a set of real data, specifically a time series of wind-speed measurements, where the clusters correspond to storms. Here the two cluster functionals selected previously correspond to the mean storm length and the mean inter-storm interval. We also consider inference for long-period return levels, advocating the posterior predictive distribution as being most representative of the information required by engineers interested in design level specifications.      

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.      

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.      

A two-dimensional inequality and uniformly continuous retractions

We prove a new inequality valid in any two-dimensional normed space. As an application, it is shown that the identity mapping on the unit ball of an infinite-dimensional uniformly convex Banach space is the mean of n uniformly continuous retractions from the unit ball onto the unit sphere, for every n?3. This last result allows us to study the extremal structure of uniformly continuous function spaces valued in an infinite-dimensional uniformly convex Banach space.     

Extreme value properties of multivariate t copulas

The extremal dependence behavior of t copulas is examined and their extreme value limiting copulas, called the t-EV copulas, are derived explicitly using tail dependence functions. As two special cases, the Hüsler–Reiss and the Marshall–Olkin distributions emerge as limits of the t-EV copula as the degrees of freedom go to infinity and zero respectively. The t copula and its extremal variants attain a wide range in the set of bivariate tail dependence parameters. Supported by NSERC Discovery Grant.     

The sum of the distances between the leaves of a tree and the 'semi-regular' property

Various topological indices have been put forward in different studies, from biochemistry to pure mathematics. Among them, the Wiener index, the number of subtrees, and the Randi? index have received great attention from mathematicians. In the study of extremal problems regarding these indices among trees, one interesting phenomenon is that they share the same extremal tree structures. Much effort was devoted to the study of the correlations between these various indices. In this note we provide a common characteristic (the ‘semi-regular’ property) of these extremal structures, with respect to the above mentioned indices, among trees with a given maximum degree. This observation leads to a more unified approach for characterizing these extremal structures. As an application/example, we illustrate the idea by studying the extremal trees, regarding the sum of distances between all pairs of leaves of a tree, a new index, which recently appeared in phylogenetic tree reconstruction, and the study of the neighborhood of trees.     

三圈图的极小广义和连通指数

图的广义和连通指数作为新提出的一类分子拓扑指数, 在QSPR/QSAR 中有很大的应用价值. 树图、单圈图和双圈图的极值问题已取得很多结果, 而三圈图相关问题的研究较为复杂. 限制 - 1\leqslant \alpha < 0, 对三圈图的广义和连通指数进行了研究. 通过对三圈图的分析, 构造了一种图的变换, 指出在三圈图中广义和连通指 数的极小值必由其中的七种类型图取得. 然后通过悬挂边的变换, 最终得到三圈图广义和连通指 数的极小值并刻画了唯一的极图.     

Extremal indices, geometric ergodicity of Markov chains, and MCMC

We investigate the connections between extremal indices on the one hand and stability of Markov chains on the other hand. Both theories relate to the tail behaviour of stochastic processes, and we find a close link between the extremal index and geometric ergodicity. Our results are illustrated throughout with examples from simple MCMC chains.      

Estimation of the extreme value index and extreme quantiles under random censoring

In this paper, we consider the estimation of the extreme value index and extreme quantiles in the presence of random right censoring. The generalization of the peaks over threshold method is discussed and an adaptation of the moment estimator is proposed. The corresponding extreme quantile estimators are also introduced. We make a start with the analysis of the asymptotic properties of the moment estimator and the corresponding extreme quantile estimator. The finite sample behaviour is illustrated with a small simulation study and through practical examples from survival data analysis.      

单圈共轭图的最小广义Randic指标R-1

图G的广义Randic指标定义为Rα=Rα（G）=∑uv∈E（G）（d（u）d（v））^α,其中d（u）是G的顶点u的度,α是任意实数．本文确定了单圈共轭图的广义Randic指标R-1的严格下界,并刻划了达到最小R-1的极图,这类极图还是化学图．     

The analysis of correlation integrals in terms of extremal value theory

In this paper we first give an overview of the methods of analysis of time series in terms of correlation integrals, which were developed for time series generated by deterministic systems. From the extremal value theory one obtains asymptotic information on the behaviour of the correlation integrals of time series generated by non-deterministic (mixing) systems. This leads to an analysis in terms of correlation integrals which is complementary to the estimation of dimension and entropy.Dedicated to the memory of Ricardo Mañé     

Tail dependence functions and vine copulas

Tail dependence and conditional tail dependence functions describe, respectively, the tail probabilities and conditional tail probabilities of a copula at various relative scales. The properties as well as the interplay of these two functions are established based upon their homogeneous structures. The extremal dependence of a copula, as described by its extreme value copulas, is shown to be completely determined by its tail dependence functions. For a vine copula built from a set of bivariate copulas, its tail dependence function can be expressed recursively by the tail dependence and conditional tail dependence functions of lower-dimensional margins. The effect of tail dependence of bivariate linking copulas on that of a vine copula is also investigated.     

单圈图的hyper-Wiener指标

令u（n）表示具有n个顶点的单圈图.在一个圈C3的一个顶点上悬挂n-3个悬挂边的n个顶点的单圈图记为U~＊（n-3,0,0）.本文证明了在u（n）中具有最小hyper-Wiener指数的单圈图是U~＊（n-3,0,0）.     

k-分支星状树的第二几何算术指标

数学化学中,第二几何算术指标是新近提出的一个图的拓扑指标,它与Szeged指标和点PI指标具有紧密关系.如果树的一个顶点υ的度大于等于3,则称顶点υ是其一个分支点.通过树的第二几何算术指标的一个增加或减少的变换,刻画了k-分支星状树的第二几何算术指标的最值,同时确定了相应的极图.     

Existence and properties of solutions of a control system with hysteresis effect

We consider a control system described by two ordinary nonlinear differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values, which depends on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input-output relation of hysteresis type.Along with the original control constraint, we also consider the convexified control constraint and the constraint consisting of the extremal points of the convexified control constraint.We prove the existence of solutions of our control system with various control constraints and establish certain relationships between corresponding solution sets.     

Distributionally robust inference for extreme Value-at-Risk

Under general multivariate regular variation conditions, the extreme Value-at-Risk of a portfolio can be expressed as an integral of a known kernel with respect to a generally unknown spectral measure supported on the unit simplex. The estimation of the spectral measure is challenging in practice and virtually impossible in high dimensions. This motivates the problem studied in this work, which is to find universal lower and upper bounds of the extreme Value-at-Risk under practically estimable constraints. That is, we study the infimum and supremum of the extreme Value-at-Risk functional, over the infinite dimensional space of all possible spectral measures that meet a finite set of constraints. We focus on extremal coefficient constraints, which are popular and easy to interpret in practice. Our contributions are twofold. First, we show that optimization problems over an infinite dimensional space of spectral measures are in fact dual problems to linear semi-infinite programs (LSIPs) – linear optimization problems in Euclidean space with an uncountable set of linear constraints. This allows us to prove that the optimal solutions are in fact attained by discrete spectral measures supported on finitely many atoms. Second, in the case of balanced portfolia, we establish further structural results for the lower bounds as well as closed form solutions for both the lower- and upper-bounds of extreme Value-at-Risk in the special case of a single extremal coefficient constraint. The solutions unveil important connections to the Tawn–Molchanov max-stable models. The results are illustrated with two applications: a real data example and closed-form formulae in a market plus sectors framework.     

Orthogonal polynomials and quadratic extremal problems

The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.

     

On extremal behaviors of Murty's least index method

In this small note, we observe some extremal behaviors of Murty's least index method for solving linear complementarity problems. In particular, we show that the expected number of steps for solving Murty's exponential example with a random permutation of variable indices is exactly equal ton, wheren is the size of the input square matrix.Corresponding author.     

On the Reverse Wiener Indices of Unicyclic Graphs

We determine the maximum values of the reverse Wiener indices of the unicyclic graphs with given cycle length, number of pendent vertices and maximum degree, respectively, and characterize the extremal graphs. We also determine the unicyclic graphs of given cycle length and diameter with minimum Wiener index.      

A Class of Trees and its Wiener Index

In this paper, we will consider the Wiener index for a class of trees that is connected to partitions of integers. Our main theorem is the fact that every integer is the Wiener index of a member of this class. As a consequence, this proves a conjecture of Lepović and Gutman. The paper also contains extremal and average results on the Wiener index of the studied class.This work was supported by Austrian Science Fund project no. S-8307-MAT.