...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/distributions/extreme.hpp>
template <class RealType = double, class Policy = policies::policy<> > class extreme_value_distribution; typedef extreme_value_distribution<> extreme_value; template <class RealType, class Policy> class extreme_value_distribution { public: typedef RealType value_type; extreme_value_distribution(RealType location = 0, RealType scale = 1); RealType scale()const; RealType location()const; };
There are various extreme value distributions : this implementation represents the maximum case, and is variously known as a FisherTippett distribution, a logWeibull distribution or a Gumbel distribution.
Extreme value theory is important for assessing risk for highly unusual events, such as 100year floods.
More information can be found on the NIST, Wikipedia, Mathworld, and Extreme value theory websites.
The relationship of the types of extreme value distributions, of which this is but one, is discussed by Extreme Value Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah.
The distribution has a PDF given by:
f(x) = (1/scale) e^{(xlocation)/scale} e^{e(xlocation)/scale}
which in the standard case (scale = 1, location = 0) reduces to:
f(x) = e^{x}e^{ex}
The following graph illustrates how the PDF varies with the location parameter:
And this graph illustrates how the PDF varies with the shape parameter:
extreme_value_distribution(RealType location = 0, RealType scale = 1);
Constructs an Extreme Value distribution with the specified location and scale parameters.
Requires scale >
0
, otherwise calls domain_error.
RealType location()const;
Returns the location parameter of the distribution.
RealType scale()const;
Returns the scale parameter of the distribution.
All the usual nonmember accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
The domain of the random parameter is [∞, +∞].
The extreme value distribution is implemented in terms of the standard
library exp
and log
functions and as such should have
very low error rates.
In the following table: a is the location parameter, b is the scale parameter, x is the random variate, p is the probability and q = 1p.
Function 
Implementation Notes 


Using the relation: pdf = exp((ax)/b) * exp(exp((ax)/b)) / b 
cdf 
Using the relation: p = exp(exp((ax)/b)) 
cdf complement 
Using the relation: q = expm1(exp((ax)/b)) 
quantile 
Using the relation: a  log(log(p)) * b 
quantile from the complement 
Using the relation: a  log(log1p(q)) * b 
mean 
a + EulerMascheroniconstant * b 
standard deviation 
pi * b / sqrt(6) 
mode 
The same as the location parameter a. 
skewness 
12 * sqrt(6) * zeta(3) / pi^{3} 
kurtosis 
27 / 5 
kurtosis excess 
kurtosis  3 or 12 / 5 