Fisher-Tippett-Gnedenko Theorem: Generalizing Three Types of Extreme Value Distributions
Fisher-Tippett-Gnedenko Theorem: Generalizing Three Types of Extreme Value Distributions
The extreme value theorem (EVT) in statistics is an analog of the central limit theorem (CLT). The idea of the CLT is that the average of many independently and identically distributed (iid) random variables converges to a normal distribution provided that each random variable has finite mean and variance. Instead of dealing with the central tendency (the mean of the parent distribution), EVT deals with the tailed region of the parent distribution.
Put simply, EVT expresses the idea that the maximum of iid random variables drawn from a parent distribution converges (in the limit of a very large sample) to one of the three possible tailed distributions, depending on the tailed index of the parent distribution: type I Gumbel distribution (=0), type II Frechet distribution (>0), or type III reversed Weibull distribution (<0).
ξ
0
ξ
0
ξ
0
ξ
0
The boundary of the tailed distribution depends on the tail index of the parent distribution. Thus, a light-tailed (=0) parent distribution places no bound on the maxima (tailed) distribution (type I Gumbel). A heavy-tailed (>0) parent puts a lower bound on the tail, whereas the lightest-tailed (<0) parent imposes an upper bound on the tail (type III reversed Weibull).
ξ
0
ξ
0
ξ
0