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:
![](I/ERMFormulaBookExtremeValueTheory_files/image006.png)
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 ![](I/ERMFormulaBookExtremeValueTheory_files/image013.png)
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
![](I/ERMFormulaBookExtremeValueTheory_files/image025.png)
if and only if that ![](I/ERMFormulaBookExtremeValueTheory_files/image020.png)
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:
![](I/ERMFormulaBookExtremeValueTheory_files/image031.png)
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