GenExtreme#

class pymc_experimental.distributions.GenExtreme(name: str, *args, rng=None, dims: str | Sequence[str | None] | None = None, initval=None, observed=None, total_size=None, transform=UNSET, **kwargs)[source]#

Univariate Generalized Extreme Value log-likelihood

The cdf of this distribution is

\[G(x \mid \mu, \sigma, \xi) = \exp\left[ -\left(1 + \xi z\right)^{-\frac{1}{\xi}} \right]\]

where

\[z = \frac{x - \mu}{\sigma}\]

and is defined on the set:

\[\left\{x: 1 + \xi\left(\frac{x-\mu}{\sigma}\right) > 0 \right\}.\]

Note that this parametrization is per Coles (2001), and differs from that of Scipy in the sign of the shape parameter, \(\xi\).

(Source code, png, hires.png, pdf)

../_images/pymc_experimental-distributions-GenExtreme-1.png

Support

  • \(x \in [\mu - \sigma/\xi, +\infty]\), when \(\xi > 0\)

  • \(x \in \mathbb{R}\) when \(\xi = 0\)

  • \(x \in [-\infty, \mu - \sigma/\xi]\), when \(\xi < 0\)

Mean

  • \(\mu + \sigma(g_1 - 1)/\xi\), when \(\xi \neq 0, \xi < 1\)

  • \(\mu + \sigma \gamma\), when \(\xi = 0\)

  • \(\infty\), when \(\xi \geq 1\) where \(\gamma\) is the Euler-Mascheroni constant, and \(g_k = \Gamma (1-k\xi)\)

Variance

  • \(\sigma^2 (g_2 - g_1^2)/\xi^2\), when \(\xi \neq 0, \xi < 0.5\)

  • \(\frac{\pi^2}{6} \sigma^2\), when \(\xi = 0\)

  • \(\infty\), when \(\xi \geq 0.5\)
Parameters:

  • mu (float) – Location parameter.

  • sigma (float) – Scale parameter (sigma > 0).

  • xi (float) – Shape parameter

  • scipy (bool) – Whether or not to use the Scipy interpretation of the shape parameter (defaults to False).

References

[Coles2001]

Coles, S.G. (2001). An Introduction to the Statistical Modeling of Extreme Values Springer-Verlag, London

__init__()#

Methods

__init__()

dist([mu, sigma, xi, scipy])

Creates a tensor variable corresponding to the cls distribution.

logcdf(mu, sigma, xi)

Compute the log of the cumulative distribution function for Generalized Extreme Value distribution at the specified value.

logp(mu, sigma, xi)

Calculate log-probability of Generalized Extreme Value distribution at specified value.

support_point(size, mu, sigma, xi)

Using the mode, as the mean can be infinite when \(\xi > 1\)

Attributes

rv_op