John H. J. Einmahl

Weighted Approximations of Tail Copula Processes with Application to Testing the Multivariate Extreme Value Condition

Consider n i.i.d. random vectors on R2, with unknown, common distribution function F.Under a sharpening of the extreme value condition on F, we derive a weighted approximation of the corresponding tail copula process.Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution.We derive the limiting distribution of the test statistic from the aforementioned weighted approximation.This limiting distribution contains unknown functional parameters.Therefore we show that a version with estimated parameters converges weakly to the true limiting distribution.Based on this result, the finite sample properties of our testing procedure are investigated through a simulation study.A real data application is also presented.

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 minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts. The minimization 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.

The almost sure behavior of maximal and minimal multivariate k n -spacings

Substrates are disclosed for use in forming printed circuit boards by the additive process, in which the surface of the substrate is contacted with a particular class of solvents, activated for electroless deposition of the metal thereon, and at one or more points in the process the board is heated or baked.

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.

Ultimate 100M World Records through Extreme-Value Theory

We use extreme-value theory to estimate the ultimate world records for the 100-m running, for both men and women. For this aim we collected the fastest personal best times set between January 1991 and June 2008. Estimators of the extreme-value index are based on a certain number of upper order statistics. To optimize this number of order statistics we minimize the asymptotic mean-squared error of the moment estimator. Using the thus obtained estimate for the extremevalue index, the right endpoint of the speed distribution is estimated. The corresponding time can be interpreted as the estimated ultimate world record: the best possible time that could be run in the near future. We find 9.51 seconds for the 100-m men and 10.33 seconds for the women.

Estimating Extreme Bivariate Quantile Regions

When simultaneously monitoring two possibly dependent, positive risks one is often interested in quantile regions with very small probability p. These extreme quantile regions contain hardly or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we take p < 1/n, with n the sample size. We consider quantile regions of the form {(x, y) Є (0,∞)2 : f(x, y)≤β}, where f, the joint density, is decreasing in both coordinates. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. As an illustration, we compute the estimated quantile regions for simulated data sets.

Goodness-of-fit Tests in Nonparametric Regression

Consider the nonparametric regression model Y = m(X) + e, where the function m is smooth, but unknown, and e is independent of X. We construct omnibus goodness-of-fit tests, based on n independent copies of (X, Y ), for the independence of e and X and establish asymptotic results for the proposed tests statistics. We investigate their finite sample properties through a simulation study and present an econometric application to household data. One testing procedure is based on differences of neighboring Y ’s, whereas the other one makes use of an estimator of m. The proofs are based on delicate weighted empirical process theory.

The Shorth Plot

The shorth plot is a tool to investigate probability mass concentration. It is a graphical representation of the length of the shorth, the shortest interval covering a certain fraction of the distribution, localized by forcing the intervals considered to contain a given point x. It is easy to compute, avoids bandwidth selection problems and allows scanning for local as well as for global features of the probability distribution. We prove functional central limit theorems for the empirical shorth plot. The good rate of convergence of the empirical shorth plot makes it useful already for moderate sample size.

Testing for Bivariate Spherical Symmetry

An omnibus test for spherical symmetry in R2 is proposed, employing localized empirical likelihood. The thus obtained test statistic is distri- bution-free under the null hypothesis. The asymptotic null distribution is established and critical values for typical sample sizes, as well as the asymptotic ones, are presented. In a simulation study, the good perfor- mance of the test is demonstrated. Furthermore, a real data example is presented.

Extreme Value Theory Approach to Simultaneous Monitoring and Thresholding of Multiple Risk Indicators

Risk assessments often encounter extreme settings with very few or no occurrences in reality.Inferences about risk indicators in such settings face the problem of insufficient data.Extreme value theory is particularly well suited for handling this type of problems.This paper uses a multivariate extreme value theory approach to establish thresholds for signaling levels of risk in the context of simultaneous monitoring of multiple risk indicators.The proposed threshold system is well justified in terms of extreme multivariate quantiles, and its sample estimator is shown to be consistent.As an illustration, the proposed approach is applied to developing a threshold system for monitoring airline performance measures.This threshold system assigns different risk levels to observed airline performance measures.In particular, it divides the sample space into regions with increasing levels of risk.Moreover, in the univariate case, such a thresholding technique can be used to determine a suitable cut-off point on a runway for holding short of landing aircrafts.This cut-off point is chosen to ensure a certain required level of safety when allowing simultaneous operations on two intersecting runways in order to ease air traffic congestion.