The extremal index,hitting time statistics and periodicity

The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors’ previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.     

基于Copula和极值理论的在险价值度量

针对传统孤立使用GJR模型、极值理论、Copula理论进行风险分析的不足,把GJR模型、极值理论和Copula理论有机的结合起来,给出了基于Copula和极值理论的投资组合VaR的测度方法.首先利用GJR模型刻画单个资产收益率中的自相关和异方差现象,获得近似独立同分布的新息序列,再分别应用高斯核估计的方法、极值理论拟合新息序列的分布函数的内部和两尾,利用Copula函数有效捕抓了市场之间的波动溢出效应,最后使用Monte Carlo模拟法,计算出投资组合的VaR值.实证结果表明,基于Copula和极值理论的VaR度量方法比历史模拟法更有效.     

基于Monte Carlo仿真的小概率事件评估算法稳定性研究

应用极值理论对小概率事件进行评估的过程中,运用仿真的方法突破了客观条件对评估样本数量的限制,样本选取数量与评估概率的关系以及极值风险评估算法稳定性是亟待讨论的理论难点问题.以飞行风险为研究对象,分析了对评估稳定性造成主要影响的随机性因素,建立了用于稳定性分析的综合考虑“物理特性”和“随机特性”的典型人-机系统仿真模型.根据仿真评估结果,归纳了评估概率随样本数量的变化规律和Monte Carlo仿真对评估稳定性的影响,提出了合理选取样本数量和评估次数的方法.研究内容可为提高小概率事件评估的稳定性、减小评估误差提供参考和借鉴.     

Free point processes and free extreme values

We continue here the study of free extreme values begun in Ben Arous and Voiculescu (Ann Probab 34:2037–2059, 2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu in Lecture notes in mathematics. Springer, Heidelberg, pp. 279–349, 1998; Barndorff-Nielsen and Thorbjornsen in Probab Theory Relat Fields 131:197–228, 2005). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (Ann Probab 34:2037–2059, 2006) and give the limit laws for free order statistics.     

Coupled Continuous Time Random Maxima

Continuous Time Random Maxima (CTRM) are a generalization of classical extreme value theory: Instead of observing random events at regular intervals in time, the waiting times between the events are also random variables which have arbitrary distributions. In case that the waiting times between the events have infinite mean, the limit process that appears differs from the limit process that appears in the classical case. With a continuous mapping approach, we derive a limit theorem for the case that the waiting times and the subsequent events are dependent as well as for the case that the waiting times depend on the preceding events (in this case we speak of an Overshooting Continuous Time Random Maxima, abbr. OCTRM). We get the distribution functions of the limit processes and a formula for the Laplace transform in time of the CTRM and the OCTRM limit. With this formula we have another way to calculate the distribution functions of the limit processes, namely by inversion of the Laplace transform. Moreover, we present governing equations which in our case are time fractional differential equations whose solutions are the distribution functions of our limit processes.     

From extended regular variation to regular variation with application in extreme value statistics

This paper aims to provide a study of a variety of concepts involving power behavior of eventually positive functions which, falling under the umbrella of the Theory of Regular Variation and its second order refinements, are prone to application in Extreme Value Theory. To this extent, some well-known properties shall be resumed, others will be designed with the ultimate purpose of establishing a relation between regular variation and extended regular variation of second order. As a by-product, new ways of looking at some common estimators for the extreme value index, in particular the maximum likelihood estimator, will be unveiled.     

Is human life limited or unlimited? (A discussion of the paper by Holger Rootzén and Dmitrii Zholud)

We thank Holger Rootzén and Dmitrii Zholud’s for their stimulating work, that led us to further investigate the problem of best fitting the human life span distribution. Their assertion “human life is unlimited but short” is based on their conclusion that the exponential model is the best to fit. We provide results based on standard Extreme Value approaches, the Block Maxima and Peaks-over-Threshold, and on the whole data available in the IDL database (IDL A, B validations and GRG). We verify that negative values for the extreme value index are more likely, supporting the conclusion that models with finite endpoint seem better to fit to the human life span distribution.     

On the limiting joint distribution of the extreme order statistics

The connection between extreme values and record-low values is exploited to derive simply the limiting joint distribution of the r largest order statistics. The use of this distribution in the modelling of corrosion phenomena is considered, and the extrapolation of maxima in space and time is described in this context. There has been recent emphasis on movement away from classical extreme value theory to more efficient estimation procedures. This shift is continued with the illustration of the extra precision of predicted maxima obtained from a model based on extreme order statistics over the classical extreme value approach.     

基于Bayes估计的金融风险值——VaR计算

初步研究了用Bayes估计计算金融风险值VaR,同时阐明了运用极值理论方法在Bayes估计下的金融风险值计算。并且借助统计计算方法——MCMC算法来求解参数的Bayes估计,有效的将Bayes思想融入到了VaR的计算中。用Bayes估计计算金融风险值VsR,可以帮助投资者将观测数据和自己所掌握的经验信息对VaR模型进行调整,使得vsR模型能够更准确地反映出金融市场的风险状况,据此做出更加正确的投资决策。     

Asymptotics for risk capital allocations based on Conditional Tail Expectation

An investigation of the limiting behavior of a risk capital allocation rule based on the Conditional Tail Expectation (CTE) risk measure is carried out. More specifically, with the help of general notions of Extreme Value Theory (EVT), the aforementioned risk capital allocation is shown to be asymptotically proportional to the corresponding Value-at-Risk (VaR) risk measure. The existing methodology acquired for VaR can therefore be applied to a somewhat less well-studied CTE. In the context of interest, the EVT approach is seemingly well-motivated by modern regulations, which openly strive for the excessive prudence in determining risk capitals.     

Emil J. Gumbel’s last course on the “Statistical theory of extreme values”: a conversation with Tuncel M. Yegulalp

GUMBEL. Eponym in mathematical statistics for the first type extreme value distribution and the copula that is both of extreme value and Archimedean kind. Hydrologists appreciate Emil J. Gumbel as a pioneer in promoting non-normal distributions in their field. Historians rank him among the most influential German intellectuals of the Weimar Republic. He disclosed secret societies that destabilized the Weimar Republic and used statistical methods to document political murders and to reveal a biased legal system. He was the first professor who lost his position for his political ideals and his stand against the national socialistic party, his books were banned and burned. Stripped of his nationality in 1933 he immigrated to France. In 1940 he escaped to the USA and settled in New York, where he was appointed Adjunct Professor at Columbia University, Department of Industrial Engineering, in 1953. We spoke with Tuncel M. Yegulalp –Professor emeritus of Mining Engineering at Columbia University– about Emil J. Gumbel’s last course on the “Statistical Theory of Extreme Values” back in 1964.     

Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process

We consider Stochastic Volatility processes with heavy tails and possible long memory in volatility. We study the limiting conditional distribution of future events given that some present or past event was extreme (i.e. above a level which tends to infinity). Even though extremes of stochastic volatility processes are asymptotically independent (in the sense of extreme value theory), these limiting conditional distributions differ from the i.i.d. case. We introduce estimators of these limiting conditional distributions and study their asymptotic properties. If volatility has long memory, then the rate of convergence and the limiting distribution of the centered estimators can depend on the long memory parameter (Hurst index).     

二元极值分布的独立性检验及在沪深股市分析中的应用

提出了二元极值分布的一个独立性检验统计量 T5,导出了它的渐近分布 ,得出了模拟分位点 ,并在小样本情况下 ,与其它已有的统计量进行比较 .结果说明本文给出的统计量 T5具有与似然比检验统计量 T3几乎相同的功效 ,且比其它检验方法有效 .最后对中国沪深两市的股票收盘指数的极值进行了独立性检验 ,认为具有显著的相关性 .     

上证指数收益率的极值研究

在极值理论广义极值分布模型的基础上,对上证指数日回报率的极值作了实证研究.给出了近两年间出现的极值的概率与等待时间,为风险的度量提供了量化的依据.     

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.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

The expected payoff to Internet auctions

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.      

The long‐term extreme price risk measure of portfolio in inventory financing: An application to dynamic impawn rate interval

Different from the short‐term risk measure for traditional financial assets (stocks, bonds, etc.), the key to illiquid inventory portfolio traded in the over‐the‐counter markets is to estimate the long‐term extreme price risk with time varying volatility. In this article, a new long‐term extreme price risk (value at risk and conditional value at risk) measure method for inventory portfolio and an application to dynamic impawn rate interval are proposed. To realize this, we first establish AutoRegressive Moving Average‐Exponential Generalized Autoregressive Conditional Heteroskedasticity‐Extreme Value Theory model and multivariatet‐Copula to depict the autocorrelation, fat tails, and volatility clustering of returns of inventories and the nonlinear dependence structure of inventories. Furthermore, we obtain the long‐term extreme price risk with time varying volatility via Monte Carlo simulation instead of square‐root‐of time rule. The results show that, first, benefits from risk diversification is significant; second, long‐term extreme price risk measure of inventory portfolio via Monte Carlo method outperforms the square‐root‐of time rule; the last is that the dynamic rate interval based on the long‐term price risk is superior to the crude rules of thumb in terms of reducing efficiency loss and improving risk coverage. In summary, this article provides a new quantitative framework for managing the risk of portfolio in inventory financing practice for banks constrained by risk limitation. © 2014 Wiley Periodicals, Inc. Complexity 20: 17–34, 2015     

二项-广义Pareto复合极值分布模型的统计推断

极值理论主要研究小概率、大影响的极端事件.当前,复合极值分布已经广泛应用于水文、气象、地震、保险、金融等领域.本文以极值类型定理和PBDH定理为理论依据,构建了二项-广义Pareto复合极值分布模型;使用概率加权矩方法,对所建立的复合模型推导参数估计式;利用计算机模拟,得到了Kolmogorov-Smirnov(简称KS)检验统计量的临界值.     

Extreme value estimation for discretely sampled continuous processes

In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.