Extreme value theory
On the decay of awfulness with oftenness
January 13, 2020 — June 30, 2021
In a satisfying way, it turns out that there are only so many shapes that probability densities can assume as they head off toward infinity. Extreme value theory makes this notion precise and gives us some tools to work with them. Important application: understanding the kinds of heavy tailed variables we can observe in nature.
See also densities and intensities, survival analysis.
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1 Tail limit theorems
The main result of use to our ends from EVT is the Pickands-Balkema-de Haan theorem (Balkema and de Haan 1974; Pickands III 1975).
This tells us that we can find a function
This maximal domain of attraction was introduced in the Fisher-Tippett theorem (Fisher and Tippett 1928), and is analysed in the EVT literature (e.g. Embrechts, Kluppelberg, and Mikosch 1997). It is pretty hard to find a distribution that does not fit in the MDA. I should try.
Practically this means that for many purposes, the tails of a random variable
2 Generalized Pareto Distribution
Best intro from Hosking and Wallis (1987):
The generalized Pareto distribution is the distribution of a random variable
defined by where is a random variable with the standard exponential distribution. The generalized Pareto distribution has distribution function
and density function
the range of is for and for The parameters of the distribution are the scale parameter, and the shape parameter. The special cases and yield, respectively, the exponential distribution with mean and the uniform distribution on Pareto distributions are obtained when
and
- The failure rate
is given by and is monotonic in decreasing if constant if and increasing if - If the random variable
has a generalized Pareto distribution, then the conditional distribution of given is also generalized Pareto, with the same value of - Let
where the are independent and identically distributed as (1) and has a Poisson distribution. Then has, essentially, a generalized extreme value (GEV) distribution as defined by Jenkinson (1955); that is, there exist quantities and independent of such that
furthermore, that is, the shape parameters of the GEV and the GPD are equal.
3 Generalized Extreme Value distributions
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4 Burr distribution
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