Annual Report 2008.pdf - SAMSI

Under asymptotic independence the largest values do not occur in the same observation, so the
region where variables are simultaneously large may not be of primary interest. A different
philosophy was offered in the paper of \cite{heffernan:tawn:2004} which allows examination of
distributional tails other than the joint tail. This approach used an asymptotic argument which
conditions on one component of the random vector and finds the limiting conditional distribution
of the remaining components as the conditioning variable becomes large. Heffernan & Resnick
(2006) provided a thorough mathematical for this conditioned limit theory. We continue to
explore this theory by considering transformation to the standard case and offer an explanation of
when this can be reduced to regular variation on a reduced cone.
Holger Rootzen
Chalmers University of Technology
rootzen@math.chalmers.se
“Three Extreme Challenges: Wind Storms, Material Fatigue, and Corrosion”
Key words: Wind storms, inclusions, stereology, pitting corrosion, design of experiments,
extreme values
We have only seen the beginning of the use of Extreme Value Statistics to contribute to solving
societal problems. It is already included in the methodological basis for several parts of
geostatistics and economic risk management, and the range of use will widen. In the future
extreme value methods will be a much more standard part of the applied statistician's toolbox.
This talk illustrates the potential for research with examples from three areas.
Wind Storms: A Weibull distribution fitted to all available data is often used to predict extreme
winds. The most extreme values then, however, have little influence on the estimated parent
Weibull distribution, and the accuracy of predictions obtained in this manner may be questioned.
We compare a Weibull method, which has been used by the Swedish meteorological office, to an
extreme value model for annual maximum wind speeds. The comparison is based on 30 years of
10 minute wind speed averages measured hourly at 12 meteorological stations located at airports
in Sweden. Results show that the Weibull method generates incorrect estimates of the tails of the
distributions of wind speeds and of the distribution of yearly maximum wind speed, and that
serial dependence has to be taken into account. The annual maxima method avoids these
problems. The measurements were rounded, first to entire knots, and then to m/s. If rounding is
disregarded then the computed standard errors of the parameter estimates become erroneously
low. Hence rounding, if done, should be taken into account in the estimation procedure. These
results are just a small corner of a classical area for Extreme Value Statistics. Much more work
has been done by, among others, Coles, Tawn, Smith and coworkers. An important challenge is
multivariate methods for wind storms.
Material Fatigue: The sizes of large inclusions within a cast of hard steel have a major influence
on fatigue characteristics, but are not directly measurable by routine means. Area Maxima or
Threshold Exceedance methods can be applied to the size distribution of large inclusions on the
basis of measurements made on a polished plane surface of the material. The two methods, of
course, are closely related and properties found by each may be deduced from those found by the