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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: 