TERStatistics of Extremes.pdf

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Emil J. Gumbel says “It’s impossible that the improbable will never happen”. Rare
events are events that have a low probability of occurrence but even if they are discrete occurrences that are statistically “improbable” in that they are very infrequent,
such events are plausible. Rare events are everywhere and encompass natural phenomena (earthquakes, tsunamis, hurricanes,. . . ), financial market crashes, etc. They
are events which recede strongly to the mean or the usual trend and which have consequences that are disastrous to human beings and environment.
Studying the occurrence of these events is of prime importance today for insurance and
financiers. Measurement and risk management have become major issues for financial
market operators, actuaries, etc.
The question is how to model the rare phenomena that lie outside the range of
available observations? Extreme value theory provides a good theoretical foundation
on which we can build statistical models describing extreme events. This modeling
corresponds to the study of the tail of the distribution that will enable us to calculate
or estimate rather extreme quantiles. Indeed, we would like to be able to answer to
these two complementary questions: Calculate the threshold zp which will be exceeded
only with a very small probability p (i.e. the (1 − p)V aR) and what is the probability
p that a disaster exceeds a threshold zp . This is the purpose of our study.
The outline of the present report is as follows. First, we give or recall some definitions and theorems. Then, extreme value theory is introduced and we see how
to estimate the extreme value index and others parameters. Finally, we present two
methods to estimate extreme quantiles: the block maxima method and the peaks-overthreshold method. All these approaches are then applied to a real dataset.