LO 45.3: Describe the peaks-over-threshold (POT) approach.

LO 45.3: Describe the peaks-over-threshold (POT) approach.
The peaks-over-threshold (POT) approach is an application of extreme value theory to the distribution of excess losses over a high threshold. The POT approach generally requires fewer parameters than approaches based on extreme value theorems. The POT approach provides the natural way to model values that are greater than a high threshold, and in this way, it corresponds to the GEV theory by modeling the maxima or minima of a large sample.
The POT approach begins by defining a random variable X to be the loss. We define u as the threshold value for positive values of x, and the distribution of excess losses over our threshold u as:
Fu (x) = P{X u u} = F(x + u) F(u) – F(u) 1 – F(u)
This is the conditional distribution for X given that the threshold is exceeded by no more than x. The parent distribution of X can be normal or lognormal, however, it will usually be unknown.
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