58 research outputs found
Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution
In this paper we perform an analytical and numerical study of Extreme Value
distributions in discrete dynamical systems. In this setting, recent works have
shown how to get a statistics of extremes in agreement with the classical
Extreme Value Theory. We pursue these investigations by giving analytical
expressions of Extreme Value distribution parameters for maps that have an
absolutely continuous invariant measure. We compare these analytical results
with numerical experiments in which we study the convergence to limiting
distributions using the so called block-maxima approach, pointing out in which
cases we obtain robust estimation of parameters. In regular maps for which
mixing properties do not hold, we show that the fitting procedure to the
classical Extreme Value Distribution fails, as expected. However, we obtain an
empirical distribution that can be explained starting from a different
observable function for which Nicolis et al. [2006] have found analytical
results.Comment: 34 pages, 7 figures; Journal of Statistical Physics 201
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
A regional Bayesian POT model for flood frequency analysis
Flood frequency analysis is usually based on the fitting of an extreme value
distribution to the local streamflow series. However, when the local data
series is short, frequency analysis results become unreliable. Regional
frequency analysis is a convenient way to reduce the estimation uncertainty. In
this work, we propose a regional Bayesian model for short record length sites.
This model is less restrictive than the index flood model while preserving the
formalism of "homogeneous regions". The performance of the proposed model is
assessed on a set of gauging stations in France. The accuracy of quantile
estimates as a function of the degree of homogeneity of the pooling group is
also analysed. The results indicate that the regional Bayesian model
outperforms the index flood model and local estimators. Furthermore, it seems
that working with relatively large and homogeneous regions may lead to more
accurate results than working with smaller and highly homogeneous regions
Universal behavior of extreme value statistics for selected observables of dynamical systems
The main results of the extreme value theory developed for the investigation
of the observables of dynamical systems rely, up to now, on the Gnedenko
approach. In this framework, extremes are basically identified with the block
maxima of the time series of the chosen observable, in the limit of infinitely
long blocks. It has been proved that, assuming suitable mixing conditions for
the underlying dynamical systems, the extremes of a specific class of
observables are distributed according to the so called Generalized Extreme
Value (GEV) distribution. Direct calculations show that in the case of
quasi-periodic dynamics the block maxima are not distributed according to the
GEV distribution. In this paper we show that, in order to obtain a universal
behaviour of the extremes, the requirement of a mixing dynamics can be relaxed
if the Pareto approach is used, based upon considering the exceedances over a
given threshold. Requiring that the invariant measure locally scales with a
well defined exponent - the local dimension -, we show that the limiting
distribution for the exceedances of the observables previously studied with the
Gnedenko approach is a Generalized Pareto distribution where the parameters
depends only on the local dimensions and the value of the threshold. This
result allows to extend the extreme value theory for dynamical systems to the
case of regular motions. We also provide connections with the results obtained
with the Gnedenko approach. In order to provide further support to our
findings, we present the results of numerical experiments carried out
considering the well-known Chirikov standard map.Comment: 7 pages, 1 figur
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