Enterprise Risk Management Formula Book
11. Extreme value theory
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See also here.
11.1 Maximum domain of attraction (MDA)
Suppose that i.i.d. random variables have cdf . Suppose also
that there exist sequences and and a cdf such that:
where is the random
variable corresponding to the block maximum for blocks of such variables of
length , i.e. each
(independent) realisation of the series is used to
create a realisation of given by .
Then is said to be
in the maximum domain of attraction (MDA) of ,
written
11.2 Fisher-Tippett theorem
If where is
a non-degenerate cdf then must be a
Generalised Extreme Value (GEV) distribution.
If where then by
replacing by and by we see that where .
11.3 The Pickands-Balkema-de Haan (PBH) theorem
Let be the
maximum limiting value of the random variable .
Then the PBH theorem states that we can find a function such that
if and only if that
11.4 Estimating tail distributions
Suppose that the underlying loss distribution is in the
maximum domain of attraction of the Frechét distribution and it has a tail of
the form: for some
slowly varying , where . Then the
Hill estimator for the (upper) tail index, given ordered
observations, , assuming
that the (upper) tail contains entries is:
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