300 research outputs found
Non-stationarity in peaks-over-threshold river flows:a regional random effects model
Under the influence of local- and large-scale climatological processes, extreme river flow events often show long-term trends, seasonality, inter-year variability and other characteristics of temporal non-stationarity. Properly accounting for this non-stationarity is vital for making accurate predictions of future floods. In this paper, a regional model based on the generalised Pareto distribution is developed for peaks-over-threshold river flow data sets when the event sizes are non-stationary. If observations are non-stationary and covariates are available then extreme value (semi-)parametric regression models may be used. Unfortunately the necessary covariates are rarely observed and, if they are, it is often not clear which process, or combination of processes, to include in the model. Within the statistical literature, latent process (or random effects) models are often used in such scenarios. We develop a regional time-varying random effects model which allows identification of temporal non-stationarity in event sizes by pooling information across all sites in a spatially homogeneous region. The proposed model, which is an instance of a Bayesian hierarchical model, can be used to predict both unconditional extreme events such as the m-year maximum, as well as extreme events that condition on being in a given year. The estimated random effects may also tell us about likely candidates for the climatological processes which cause non-stationarity in the flood process. The model is applied to UK flood data from 817 stations spread across 81 hydrometric regions
Functional kernel estimators of conditional extreme quantiles
We address the estimation of "extreme" conditional quantiles i.e. when their
order converges to one as the sample size increases. Conditions on the rate of
convergence of their order to one are provided to obtain asymptotically
Gaussian distributed kernel estimators. A Weissman-type estimator and kernel
estimators of the conditional tail-index are derived, permitting to estimate
extreme conditional quantiles of arbitrary order.Comment: arXiv admin note: text overlap with arXiv:1107.226
Generalized Extreme Value distribution parameters as dynamical indicators of Stability
We introduce a new dynamical indicator of stability based on the Extreme
Value statistics showing that it provides an insight on the local stability
properties of dynamical systems. The indicator perform faster than other based
on the iteration of the tangent map since it requires only the evolution of the
original systems and, in the chaotic regions, gives further information about
the information dimension of the attractor. A numerical validation of the
method is presented through the analysis of the motions in a Standard map
Bias reduction in traceroute sampling: towards a more accurate map of the Internet
Traceroute sampling is an important technique in exploring the internet
router graph and the autonomous system graph. Although it is one of the primary
techniques used in calculating statistics about the internet, it can introduce
bias that corrupts these estimates. This paper reports on a theoretical and
experimental investigation of a new technique to reduce the bias of traceroute
sampling when estimating the degree distribution. We develop a new estimator
for the degree of a node in a traceroute-sampled graph; validate the estimator
theoretically in Erdos-Renyi graphs and, through computer experiments, for a
wider range of graphs; and apply it to produce a new picture of the degree
distribution of the autonomous system graph.Comment: 12 pages, 3 figure
Using Extreme Value Theory for Determining the Probability of Carrington-Like Solar Flares
Space weather events can negatively affect satellites, the electricity grid,
satellite navigation systems and human health. As a consequence, extreme space
weather has been added to the UK and other national risk registers. By their
very nature, extreme space weather events occur rarely and, therefore,
statistical methods are required to determine the probability of their
occurrence. Space weather events can be characterised by a number of natural
phenomena such as X-ray (solar) flares, solar energetic particle (SEP) fluxes,
coronal mass ejections and various geophysical indices (Dst, Kp, F10.7). In
this paper extreme value theory (EVT) is used to investigate the probability of
extreme solar flares. Previous work has assumed that the distribution of solar
flares follows a power law. However such an approach can lead to a poor
estimation of the return times of such events due to uncertainties in the tails
of the probability distribution function. Using EVT and GOES X-ray flux data it
is shown that the expected 150-year return level is approximately an X60 flare
whilst a Carrington-like flare is a one in a 100-year event. It is also shown
that the EVT results are consistent with flare data from the Kepler space
telescope mission.Comment: 13 pages, 4 figures; updated content following reviewer feedbac
Yet another breakdown point notion: EFSBP - illustrated at scale-shape models
The breakdown point in its different variants is one of the central notions
to quantify the global robustness of a procedure. We propose a simple
supplementary variant which is useful in situations where we have no obvious or
only partial equivariance: Extending the Donoho and Huber(1983) Finite Sample
Breakdown Point, we propose the Expected Finite Sample Breakdown Point to
produce less configuration-dependent values while still preserving the finite
sample aspect of the former definition. We apply this notion for joint
estimation of scale and shape (with only scale-equivariance available),
exemplified for generalized Pareto, generalized extreme value, Weibull, and
Gamma distributions. In these settings, we are interested in highly-robust,
easy-to-compute initial estimators; to this end we study Pickands-type and
Location-Dispersion-type estimators and compute their respective breakdown
points.Comment: 21 pages, 4 figure
Spectral Density Ratio Models for Multivariate Extremes
The modeling of multivariate extremes has received increasing recent attention because of its importance in risk assessment. In classical statistics of extremes, the joint distribution of two or more extremes has a nonparametric form, subject to moment constraints. This article develops a semiparametric model for the situation where several multivariate extremal distributions are linked through the action of a covariate on an unspecified baseline distribution, through a so-called density ratio model. Theoretical and numerical aspects of empirical likelihood inference for this model are discussed, and an application is given to pairs of extreme forest temperatures. Supplementary materials for this article are available online
On the modelling of the excesses of galaxy clusters over high-mass thresholds
In this work we present for the first time an application of the Pareto
approach to the modelling of the excesses of galaxy clusters over high-mass
thresholds. The distribution of those excesses can be described by the
generalized Pareto distribution (GPD), which is closely related to the
generalized extreme value (GEV) distribution. After introducing the formalism,
we study the impact of different thresholds and redshift ranges on the
distributions, as well as the influence of the survey area on the mean excess
above a given mass threshold. We also show that both the GPD and the GEV
approach lead to identical results for rare, thus high-mass and high-redshift,
clusters. As an example, we apply the Pareto approach to ACT-CL J0102-4915 and
SPT-CL J2106-5844 and derive the respective cumulative distribution functions
of the exceedance over different mass thresholds. We also study the possibility
to use the GPD as a cosmological probe. Since in the maximum likelihood
estimation of the distribution parameters all the information from clusters
above the mass threshold is used, the GPD might offer an interesting
alternative to GEV-based methods that use only the maxima in patches. When
comparing the accuracy with which the parameters can be estimated, it turns out
that the patch-based modelling of maxima is superior to the Pareto approach. In
an ideal case, the GEV approach is capable to estimate the location parameter
with a percent level precision for less than 100 patches. This result makes the
GEV based approach potentially also interesting for cluster surveys with a
smaller area.Comment: 10 pages, 8 figures, MNRAS accepted, minor modifications to match the
accepted versio
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
A preferential attachment model with random initial degrees
In this paper, a random graph process is studied and its
degree sequence is analyzed. Let be an i.i.d. sequence. The
graph process is defined so that, at each integer time , a new vertex, with
edges attached to it, is added to the graph. The new edges added at time
t are then preferentially connected to older vertices, i.e., conditionally on
, the probability that a given edge is connected to vertex i is
proportional to , where is the degree of vertex
at time , independently of the other edges. The main result is that the
asymptotical degree sequence for this process is a power law with exponent
, where is the power-law exponent
of the initial degrees and the exponent predicted
by pure preferential attachment. This result extends previous work by Cooper
and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is
incomplete. This version contains the complete proo
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