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### Enterprise Risk Management Formula Book 11. Extreme value theory

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: