Multivariate Extreme Value Research Articles

Accounting for uncertainty in extremal dependence modeling using Bayesian model averaging techniques

Modeling the joint tail of an unknown multivariate distribution can be characterized as modeling the tail of each marginal distribution and modeling the dependence structure between the margins. Classical methods for modeling multivariate extremes are based on the class of multivariate extreme value distributions. However, such distributions do not allow for the possibility of dependence at finite levels that vanishes in the limit. Alternative models have been developed that account for this asymptotic independence, but inferential statistical procedures seeking to combine the classes of asymptotically dependent and asymptotically independent models have been of limited use. We overcome these difficulties by employing Bayesian model averaging to account for both types of asymptotic behavior, and for subclasses within the asymptotically independent framework. Our approach also allows for the calculation of posterior probabilities of different classes of models, allowing for direct comparison between them. We demonstrate the use of joint tail models based on our broader methodology using two oceanographic datasets and a brief simulation study.

Early Warning Systems for Currency Crises: A Multivariate Extreme Value Approach

We apply extreme value theory to assess the tail dependence between three currency crises measures and 18 economic indicators commonly used for predicting crises. In our pooled sample of 46 countries in the period 1974-2008, we find that nearly all pairs of variables are asymptotically independent: in the limit, extreme values of economic indicators are not followed by severe currency crashes. Our findings may explain the poor performance of existing early warning systems for currency crises. However, we do find that economic variables with stronger extremal association with the exchange rate have better crisis prediction performance, both in-sample and out-of-sample.

Multivariate multiparameter extreme value models and return periods: A copula approach

Multivariate extreme value models are a fundamental tool in order to assess potentially dangerous events. The target of this paper is two‐fold. On the one hand we outline how, exploiting recent theoretical developments in the theory of copulas, new multivariate extreme value distributions can be easily constructed; in particular, we show how a suitable number of parameters can be introduced, a feature not shared by traditional extreme value models. On the other hand, we introduce a proper new definition of multivariate return period and show the differences with (and the advantages over) the definition presently used in literature. An illustration involving flood data is presented and discussed, and a generalization of the well‐known multivariate logistic Gumbel model is also given.

Novelty Detection with Multivariate Extreme Value Statistics

Novelty detection, or one-class classification, aims to determine if data are "normal" with respect to some model of normality constructed using examples of normal system behaviour. If that model is composed of generative probability distributions, the extent of "normality" in the data space can be described using Extreme Value Theory (EVT), a branch of statistics concerned with describing the tails of distributions. This paper demonstrates that existing approaches to the use of EVT for novelty detection are appropriate only for univariate, unimodal problems. We generalise the use of EVT for novelty detection to the analysis of data with multivariate, multimodal distributions, allowing a principled approach to the analysis of high-dimensional data to be taken. Examples are provided using vital-sign data obtained from a large clinical study of patients in a high-dependency hospital ward.

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.

COMOVEMENTS OF DIFFERENT ASSET CLASSES DURING MARKET STRESS

This paper assesses the linkages between the most important U.S. financial asset classes (stocks, bonds, T-bills and gold) during periods of financial turmoil. Our results have potentially important implications for strategic asset allocation and pension fund management. We use multivariate extreme value theory to estimate the exposure of one asset class to extreme movements in the other asset classes. By applying structural break tests to those measures we study to what extent linkages in extreme asset returns and volatilities are changing over time. Univariate results andch bivariate comovement results exhibit significant breaks in the 1970s and 1980s corresponding to the turbulent times of e.g. the oil shocks, Volcker’s presidency of the Fed or the stock market crash of 1987.

On Prediction of Bivariate Extremes

There are mainly two competing approaches to modeling high dimensional extremes, namely multivariate extreme value distributions and multivariate peaks over threshold models which lead to a class of distribution called multivariate generalized Pareto distributions. Although the probability theory for the latter models is fairly well developed the statistical properties of them are generally unknown. We compare performances of these models for prediction of extremes under different circumstances and apply the results to modeling of real wind speed data. When modeling such extreme events one should not leave out of consideration that observations measured in closely located stations usually show strong dependence. Thus, besides fitting univariate margins, the knowledge of the dependence structure among the stations is also crucial. For the bivariate maxima the parametric cases are fully developed, but these structures are not always fiexible enough for real applications. A promising alternative way f or modeling the dependence could be obtained by non-parametric dependence functions. The most efficient known non-parametric models were introduced by Capeera et al. [3] and Hall and Tajvidi [7]. However to obtain density estimation further refinements are needed, since these approaches do not result in dependence functions which are differentiable everywhere. To tackle this problem polynomial smoothing splines have been considered taking into account all required constraints on dependence functions. It should be noted that investigating “only” the maxima can hide the time structure within the given period, so we do not know whether the different components of the maxima occurred really simultaneously (e.g., in the same day) or not. To avoid this problem exceedances over a high threshold can be considered. We applied a new definition for describing the distribution of the exceedances proposed by Rootzen and Tajvidi [11]. The main curiosity of it is including also those observations in modeling which are above the threshold at least in one component. Both of the approaches for maxima and exceedances have been applied for bivariate datasets arising from wind time series of the last 5 decades measured in north Germany. We compute prediction regions for all fitted models, which makes the models easily comparable. Finally, as the statistical properties of the proposed exceedance model is still not fully studied, we investigate its accuracy and compare it with rather standard block maxima approach by a simulation study.

Joint modelling of wave spectral parameters for extreme sea states

Characterising the dependence between extremes of wave spectral parameters such as significant wave height ( H S ) and spectral peak period ( T P ) is important in understanding extreme ocean environments and in the design and assessment of marine structures. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity. Here we seek to characterise joint dependence in a straightforward manner, accessible to the ocean engineering community, using a statistically sound approach. Many methods of multivariate extreme value analyses are based on models which assume implicitly that in some joint tail region each parameter is either independent of or asymptotically dependent on other parameters; yet in reality the dependence structure in general is neither of these. The underpinning assumption of multivariate regular variation restricts these methods to estimation of joint regions in which all parameters are extreme; but regions where only a subset of parameters are extreme can be equally important for design. The conditional approach of Heffernan and Tawn (2004), similar in spirit to that of Haver (1985) but with better theoretical foundation, overcomes these difficulties. We use the conditional approach to characterise the dependence structure of H S and T P . The key elements of the procedure are: (1) marginal modelling for all parameters, (2) transformation of data to a common standard Gumbel marginal form, (3) modelling dependence between data for extremes of pairs of parameters using a form of regression, (4) simulation of long return periods to estimate joint extremes. We demonstrate the approach in application to measured and hindcast data from the Northern North Sea, the Gulf of Mexico and the North West Shelf of Australia. We also illustrate the use of data re-sampling techniques such as bootstrapping to estimate the uncertainty in marginal and dependence models and accommodate this uncertainty in extreme quantile estimation. We discuss the current approach in the context of other approaches to multivariate extreme value estimation popular in the ocean engineering community.

Influence measures and robust estimators of dependence in multivariate extremes

We develop a simple influence measure to assess whether Bayesian estimators in multivariate extreme value problems are sensitive to outliers. The proposed measure is easy to compute by importance sampling and successfully captures two effects on the functional: the “data effect” and the “parameter uncertainty effect”. We also propose a new Bayesian estimator which is easy to implement and is robust. The methods are tested and illustrated using simulated data and then applied to stock market data.

The pairwise beta distribution: A flexible parametric multivariate model for extremes

We present a new parametric model for the angular measure of a multivariate extreme value distribution. Unlike many parametric models that are limited to the bivariate case, the flexible model can describe the extremes of random vectors of dimension greater than two. The novel construction method relies on a geometric interpretation of the requirements of a valid angular measure. An advantage of this model is that its parameters directly affect the level of dependence between each pair of components of the random vector, and as such the parameters of the model are more interpretable than those of earlier parametric models for multivariate extremes. The model is applied to air quality data and simulated spatial data.

Dependence structure of risk factors and diversification effects

In this paper, we study the aggregated risk from dependent risk factors under the multivariate Extreme Value Theory (EVT) framework. We consider the heavy-tailedness of the risk factors as well as the non-parametric tail dependence structure. This allows a large range of models on the dependence. We assess the Value-at-Risk of a diversified portfolio constructed from dependent risk factors. Moreover, we examine the diversification effects under this setup.

Are Banks Too Big To Fail? Measuring Systemic Importance of Financial Institutions

We consider three measures on the systemic importance of a financial institution within a interconnected financial system. Based on the measures, we study the relation between the size of a financial institution and its systemic importance. From both theoretical model and empirical analysis, we find that in analyzing the systemic risk posed by one financial institution to the system, size should not be considered as a proxy of systemic importance. In other words, the too big to fail argument is not always valid, and alternative measures on systemic importance should be considered. We provide the estimation methodology of systemic importance measures under the multivariate Extreme Value Theory (EVT) framework.

On the estimation and application of max-stable processes

The theory of max-stable processes generalizes traditional univariate and multivariate extreme value theory by allowing for processes indexed by a time or space variable. We consider a particular class of max-stable processes, known as M4 processes, that are particularly well adapted to modeling the extreme behavior of multiple time series. We develop procedures for determining the order of an M4 process and for estimating the parameters. To illustrate the methods, some examples are given for modeling jumps in returns in multivariate financial time series. We introduce a new measure to quantify and predict the extreme co-movements in price returns.

Thresholding Events of Extreme in Simultaneous Monitoring of Multiple Risks

This article develops a threshold system for monitoring airline performance. This threshold system divides the sample space into regions with increasing levels of risk and allows instant assessments of risk level of any observed airline performance. Of particular concern is the performance with extreme risk. In this article, a multivariate extreme value theory approach is used to establish thresholds for signaling varying levels of extremeness in the context of simultaneous monitoring of multiple risk measures. The threshold system is justified in terms of multivariate extreme quantiles, and its sample estimator is shown to be consistent. This threshold system applies to general extreme risk management. Finally, a simulation and comparison study demonstrates the good performance of the proposed multivariate extreme quantile estimator. Supplemental materials providing technical details are available online.

Dependence Structure of Risk Factors and Diversification Effects

In this paper, we study the aggregated risk from dependent risk factors under the multivariate Extreme Value Theory (EVT) framework. We consider the heavy-tailness of the risk factors as well a non-parametric tail dependence structure. This allows a large scope of models on the dependency. We assess the Value-at-Risk of a diversified portfolio constructed from dependent risk factors. Moreover, we examine the diversification effects under this setup.

On the construction of multivariate extreme value models via copulas

AbstractCopulas represent a fundamental tool for constructing multivariate probability distributions. Exploiting recent theoretical developments concerning the construction of copulas, we outline several methods for generating multivariate extreme value (MEV) laws having a suitable number of parameters, a feature of great importance in applications. The corresponding random vectors can be efficiently simulated, and easily fitted to empirical data. The use of multivariate return periods for extreme events is also discussed. A practical illustration involving maxima sampled via a network of non‐independent gauge stations is presented. Copyright © 2009 John Wiley & Sons, Ltd.

HIGH‐DIMENSIONAL PARAMETRIC MODELLING OF MULTIVARIATE EXTREME EVENTS

SummaryMultivariate extreme events are typically modelled using multivariate extreme value distributions. Unfortunately, there exists no finite parametrization for the class of multivariate extreme value distributions. One common approach is to model extreme events using some flexible parametric subclass. This approach has been limited to only two or three dimensions, primarily because suitably flexible high‐dimensional parametric models have prohibitively complex density functions. We present an approach that allows a number of popular flexible models to be used in arbitrarily high dimensions. The approach easily handles missing and censored data, and can be employed when modelling componentwise maxima and multivariate threshold exceedances. The approach is based on a representation using conditionally independent marginal components, conditioning on positive stable random variables. We use Bayesian inference, where the conditioning variables are treated as auxiliary variables within Markov chain Monte Carlo simulations. We demonstrate these methods with an application to sea‐levels, using data collected at 10 sites on the east coast of England.

Models for Dependent Extremes Using Stable Mixtures

Abstract. This paper unifies and extends results on a class of multivariate extreme value (EV) models studied by Hougaard, Crowder and Tawn. In these models, both unconditional and conditional distributions are themselves EV distributions, and all lower‐dimensional marginals and maxima belong to the class. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential‐stable location mixtures (for Gumbel) or as power‐stable scale mixtures (for non‐Gumbel EV distributions). A third interpretation is through a peaks over thresholds model with a positive stable intensity. The mixing variables are used as a modelling tool and for better understanding and model checking. We study EV analogues of components of variance models, and new time series, spatial and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.

Expansions of multivariate Pickands densities and testing the tail dependence

Multivariate extreme value distribution functions (EVDs) with standard reverse exponential margins and the pertaining multivariate generalized Pareto distribution functions (GPDs) can be parametrized in terms of their Pickands dependence function D with D = 1 representing tail independence. Otherwise, one has to deal with tail dependence. Besides GPDs we include in our statistical model certain distribution functions (dfs) which deviate from the GPDs whereby EVDs serve as special cases. Our aim is to test tail dependence against rates of tail independence based on the radial component. For that purpose we study expansions and introduce a second order condition for the density (called Pickands density) of the joint distribution of the angular and radial component with the Pickands densities under GPDs as leading terms. A uniformly most powerful test procedure is established based on asymptotic distributions of radial components. It is argued that there is no loss of information if the angular component is omitted in the testing problem.

A New Class of Models for Bivariate Joint Tails

SummaryA fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.