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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