A method of moments estimator of tail dependence

In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ130 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Assessing Confidence Intervals for the Tail Index by Edgeworth Expansions for the Hill Estimator

AMS classifications: 62G20, 62G32;asymptotic normality;confidence intervals;Edgeworth expansions;extreme value index;Hill estimator;regular variation;tail index

Unbiased Tail Estimation by an Extension of the Generalized Pareto Distribution

AMS classifications: 62G20; 62G32;bias;exchange rate;heavy tails;peaks-over-threshold;regular variation;tail index

Mandelbrot's Extremism

In the sixties Mandelbrot already showed that extreme price swings are more likely than some of us think or incorporate in our models.A modern toolbox for analyzing such rare events can be found in the field of extreme value theory.At the core of extreme value theory lies the modelling of maxima over large blocks of observations and of excesses over high thresholds.The general validity of these models makes them suitable for out-of-sample extrapolation.By way of illustration we assess the likeliness of the crash of the Dow Jones on October 19, 1987, a loss that was more than twice as large as on any other single day from 1954 until 2004.exceedances;extreme value theory;heavy tails;maxima

An M-Estimator for Tail Dependence in Arbitrary Dimensions

Consider a random sample in the max-domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model. A new estimator for the unknown parameter is defined as the value that minimises the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts. The minimisation problem has, with probability tending to one, a unique, global solution. The estimator is consistent and asymptotically normal. The spectral measures of the tail dependence models to which the method applies can be discrete or continuous. Examples demonstrate the applicability and the performance of the method.asymptotic statistics;factor model;M-estimation;multivariate extremes;tail dependence

Projection Estimates of Constrained Functional Parameters

AMS classifications: 62G05; 62G07; 62G08; 62G20; 62G32;estimation;convex function;extreme value copula;Pickands dependence function;projection;shape constraint;support function;tangent cone

Inference on the tail process with application to financial time series modelling

To draw inference on serial extremal dependence within heavy-tailed Markov chains, Drees, Segers and Warcho{\l} [Extremes (2015) 18, 369--402] proposed nonparametric estimators of the spectral tail process. The methodology can be extended to the more general setting of a stationary, regularly varying time series. The large-sample distribution of the estimators is derived via empirical process theory for cluster functionals. The finite-sample performance of these estimators is evaluated via Monte Carlo simulations. Moreover, two different bootstrap schemes are employed which yield confidence intervals for the pre-asymptotic spectral tail process: the stationary bootstrap and the multiplier block bootstrap. The estimators are applied to stock price data to study the persistence of positive and negative shocks.Comment: 22 page

Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials

Many applications in risk analysis, especially in environmental sciences, require the estimation of the dependence among multivariate maxima. A way to do this is by inferring the Pickands dependence function of the underlying extreme-value copula. A nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of a specific type of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated by analyzing clusters consisting of seven weather stations that have recorded weekly maxima of hourly rainfall in France from 1993 to 2011

Assessing Confidence Intervals for the Tail Index by Edgeworth Expansions for the Hill Estimator

AMS classifications: 62G20, 62G32;

Formation of a Metallic Contact: Jump to Contact Revisited

The transition from tunneling to metallic contact between two surfaces does not always involve a jump, but can be smooth. We have observed that the configuration and material composition of the electrodes before contact largely determines the presence or absence of a jump. Moreover, when jumps are found preferential values of conductance have been identified. Through combination of experiments, molecular dynamics, and first-principles transport calculations these conductance values are identified with atomic contacts of either monomers, dimers or double-bond contacts.Comment: 4 pages, 5 figure