Statistics and Algorithms / Probability Distributions /

GEV (Generalized Extreme Value)

The Generalized Extreme Value distribution is a family of distributions which form the possible limiting distributions for the extreme values of a set of i.i.d. random variables.

Parameters	\(k\) is the tail or shape parameter, \(\mu\) is the location and \(\sigma\) is the scale. \(\sigma\) needs to be positive; the other two can take any value.
Kernel	\(\frac{1}{\sigma }t(x)^{k + 1} e^{ - t(x)} \) \(t(x) = \left[ {1 + k\left( {\frac{{x - \mu }}{\sigma }} \right)} \right]^{ - \frac{1}{k}} \,{\rm{if}}\,k \ne 0,\exp \left( { - \frac{{x - \mu }}{\sigma }} \right) \,{\rm{if}}\,k = 0\)
Support	Depends upon \(k\). \(\left( { - \infty ,\infty } \right)\) when \(k=0\)
Main Uses	An extreme value disturbance in the index of a choice model generates a logit model. Properly scaled can be used for handling tail probabilities.
Density Function	%loggevdensity(x,k,mu,sigma) is the log density (x\|k,mu,sigma)
Distribution Function	%gevcdf(x,k,mu,sigma) is the cumulative CDF for (x\|k,mu,sigma) %invgev(p,k,mu,sigma) is the inverse CDF for probability p given parameters k, mu and sigma.