28 research outputs found
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 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 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 -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