Extreme Value Distribution

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##### Extreme Value Distribution

    #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 Fisher-Tippett distribution, a log-Weibull distribution or a Gumbel distribution.

Extreme value theory is important for assessing risk for highly unusual events, such as 100-year floods.

More information can be found on the NIST, Wikipedia, Mathworld, and Extreme value theory websites.

The distribution has a PDF given by:

f(x) = (1/scale) e-(x-location)/scale e-e-(x-location)/scale

Which in the standard case (scale = 1, location = 0) reduces to:

f(x) = e-xe-e-x

The following graph illustrates how the PDF varies with the location parameter:

And this graph illustrates how the PDF varies with the shape parameter:

##### Member Functions

    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.

##### Non-member Accessors

All the usual non-member 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 [-∞, +∞].

##### Accuracy

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.

##### Implementation

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 = 1-p.

Function | Implementation Notes
--- | ---
pdf | Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) / b
cdf | Using the relation: p = exp(-exp((a-x)/b))
cdf complement | Using the relation: q = -expm1(-exp((a-x)/b))
quantile | Using the relation: a - log(-log(p)) * b
quantile from the complement | Using the relation: a - log(-log1p(-q)) * b
mean | a + Euler-Mascheroni-constant * b
standard deviation | pi * b / sqrt(6)
mode | The same as the location parameter a.
skewness | 12 * sqrt(6) * zeta(3) / pi3
kurtosis | 27 / 5
kurtosis excess | kurtosis - 3 or 12 / 5