GENERALIZED EXTREME VALUE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.GeneralizedExtremeValue({"xi": *, "mu": *, "sigma": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{GeneralizedExtremeValue}(\xi ,\mu ,\sigma )$$

Distribution Domain

$$\text{if }\xi >0:x\in (z(x),\mathrm{\infty }),\text{if }\xi =0:x\in (-\mathrm{\infty },\mathrm{\infty }),\text{if }\xi <0:x\in (-\mathrm{\infty },z(x))$$

Parameters Domain and Constraints

$$\xi \in \mathbb{R},\mu \in \mathbb{R},\sigma \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\{\begin{array}{cl}\exp (-\exp (-z(x)))& \text{if }\xi =0\\ \exp (-(1+\xi z(x){)}^{-1/\xi })& \text{if }\xi \ne 0\end{array}$$

Probability Density Function

$$f_X\left(x\right)=\{\begin{array}{cl}\frac{1}{\sigma }\exp (-z(x))\exp (-\exp (-z(x)))& \text{if }\xi =0\\ \frac{1}{\sigma }(1+\xi z(x){)}^{-(1+1/\xi )}\exp (-(1+\xi z(x){)}^{-1/\xi })& \text{if }\xi \ne 0\end{array}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\{\begin{array}{cl}\mu -\sigma \ln (-\ln \left(u\right))& \text{if }\xi =0\\ \mu +\frac{}{}\frac{\sigma }{\xi }({(-\ln (u))}^{-\xi }-1)& \text{if }\xi \ne 0\end{array}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx=\mathrm{\Gamma }(1-k\xi )$$

Parametric Mean

$$\mathrm{Mean}(X)=\{\begin{array}{cl}\mu +\sigma ({\mu }_1^{\prime }-1)/\xi & \text{if}\xi \ne 0,\xi <1\\ \mu +\sigma \gamma & \text{if}\xi =0\end{array}$$

Parametric Variance

$$\mathrm{Variance}(X)=\{\begin{array}{cl}{\sigma }^2({\mu }_2^{\prime }-{\mu }_1^{\prime 2})/{\xi }^2& \text{if}\xi \ne 0,\xi <\frac{1}{2}\\ {\sigma }^2\frac{{\pi }^2}{6}& \text{if}\xi =0\end{array}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\{\begin{array}{cl}\text{sign}(\xi )\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}& \text{if}\xi \ne 0,\xi <\frac{1}{3}\\ \frac{12\sqrt{6}\zeta (3)}{{\pi }^3}& \text{if}\xi =0\end{array}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\{\begin{array}{cl}3+\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}& \text{if}\xi \ne 0,\xi <\frac{1}{4}\\ 3+\frac{12}{5}& \text{if}\xi =0\end{array}$$

Parametric Median

$$\mathrm{Median}(X)=\{\begin{array}{cl}\mu +\sigma \frac{(\ln 2{)}^{-\xi }-1}{\xi }& \text{if }\xi \ne 0\\ \mu -\sigma \ln \ln 2& \text{if}\xi =0\end{array}$$

Parametric Mode

$$\mathrm{Mode}(X)=\{\begin{array}{cl}\mu +\sigma \frac{(1+\xi {)}^{-\xi }-1}{\xi }& \text{if }\xi \ne 0\\ \mu & \text{if }\xi =0\end{array}$$

Additional Information and Definitions

• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$
• $\gamma :\text{Euler-Mascheroni constant}=0.5772156649$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document