A multivariate piecing-together approach with an application to operational loss data

The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins. The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuous manner. The result is again a copula. The other step consists of the transformation of each margin of this new copula by a given univariate distribution function. This provides, altogether, a multivariate distribution function with prescribed margins whose copula coincides in its central part with $C$ and in its upper tail with a GPD copula. When applied to data, this approach also enables the evaluation of a wide range of rational scenarios for the upper tail of the underlying distribution function in the multivariate case. We apply this approach to operational loss data in order to evaluate the range of operational risk.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ343 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The multivariate Piecing-Together approach revisited

The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (2012a), we achieve in the present paper an exact representation of the PT-copula's upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.Comment: 12 pages, 1 figure. To appear in the Journal of Multivariate Analysi

Testing for a generalized Pareto process

We investigate two models for the following setup: We consider a stochastic process X \in C[0,1] whose distribution belongs to a parametric family indexed by \vartheta \in {\Theta} \subset R. In case \vartheta = 0, X is a generalized Pareto process. Based on n independent copies X(1),...,X(n) of X, we establish local asymptotic normality (LAN) of the point process of exceedances among X(1),...,X(n) above an increasing threshold line in each model. The corresponding central sequences provide asymptotically optimal sequences of tests for testing H0 : \vartheta = 0 against a sequence of alternatives Hn : \vartheta = \varthetan converging to zero as n increases. In one model, with an underlying exponential family, the central sequence is provided by the number of exceedances only, whereas in the other one the exceedances themselves contribute, too. However it turns out that, in both cases, the test statistics also depend on some additional and usually unknown model parameters. We, therefore, consider an omnibus test statistic sequence as well and compute its asymptotic relative efficiency with respect to the optimal test sequence.Comment: 22 pages, 1 figure, 4th International Conference of the ERCIM WG on Computing & Statistics (ERCIM'11), 17-19 December, 2011, University of Londo

Triangular Spin-Orbit-Coupled Lattice with Strong Coulomb Correlations: Sn Atoms on a SiC(0001) Substrate

Two-dimensional (2D) atom lattices provide model setups for Coulomb correlations inducing competing ground states, partly with topological character. Hexagonal SiC(0001) is an intriguing wide-gap substrate, spectroscopically separated from the overlayer and hence reduced screening. We report the first study of an artificial high-Z atom lattice on SiC(0001) by Sn adatoms, based on combined experimental realization and theoretical modeling. Density-functional theory of our $\sqrt{3}$-structure model closely reproduces the scanning tunneling microscopy. Instead of metallic behavior, photoemission data show a deeply gapped state (~2 eV gap). Based on our calculations including dynamic mean-field theory, we argue that this reflects a pronounced Mott insulating scenario. We also find indications that the system is susceptible to antiferromagnetic superstructures. Such spin-orbit-coupled correlated heavy atom lattices on SiC(0001) thus form a novel testbed for peculiar quantum states of matter, with potential bearing for spin liquids and topological Mott insulators.Comment: 5 pages, 4 figures, 1 tabl

On Max-Stable Processes and the Functional D-Norm

We introduce a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions (GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case. Applying this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD. \delta-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known in the multivariate case.Comment: 22 page