Extreme Value Theory - Data Sampling

Data Sampling

Two approaches exist to fit the tail of a sample empirical cumulative distribution function (ECDF) to one of the three possible distribution functions (called the Frechet, Gumbel and Weibull). The first method relies on approximating a distribution from a so-called block maxima (minima) series. In operational statistics situations, it is customary and convenient to apply a sampling method that consists in extracting the annual maxima. In doing so, a so-called "Annual Maxima Series" (AMS) is generated. The second method relies on sampling points from the data set that exceeds a certain threshold (falls below a certain floor). This method is generally referred to as the "Point Over Threshold" method (POT).:

1. Basic theory approach as described in the book by Burry (1975). In general this conforms to the first theorem in extreme value theory (Fisher and Tippett, 1928; Gnedenko, 1943).
2. Most common at this moment is the tail-fitting approach based on the second theorem in extreme value theory (Pickands, 1975; Balkema and de Haan, 1974).

The difference between the two theorems is due to the nature of the data generation. For Theorem I the data are generated in full range, while in Theorem II data is only generated when it surpasses a certain threshold, called Peak Over Threshold models (POT). The POT approach has been developed largely in the insurance business, where only losses (pay outs) above a certain threshold are accessible to the insurance company. Strangely, this approach is often used for cases where Theorem I applies, which creates problems with the basic model assumptions.

Extreme value distributions are the limiting distributions for the minimum or the maximum of a very large collection of independent random variables from the same arbitrary distribution. Emil Julius Gumbel (1958) showed that for any well-behaved initial distribution (i.e., F(x) is continuous and has an inverse), only a few models are needed, depending on whether you are interested in the maximum or the minimum, and also if the observations are bounded above or below.