GENERALIZED PARETO DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.GeneralizedPareto({"c": *, "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

$$X\sim \mathrm{GeneralizedPareto}(c,\mu ,\sigma )$$

Distribution Domain

$$\text{if }c⩾0:x\in (\mu ,\mathrm{\infty }),\text{if }c<0:x\in (-\mathrm{\infty },\mu -\frac{\sigma }{c})$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=1-(1+cz(x){)}^{-1/c}$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\sigma }(1+cz(x){)}^{-(1/c+1)}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\mu +\frac{\sigma (u^{-c}-1)}{c}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx=\frac{{(-1)}^k}{c^k}\sum _{i=0}^k(\genfrac{}{}{0ex}{}{k}{i})\frac{{(-1)}^i}{1-ci}\text{if }<\frac{1}{k}$$

Parametric Mean

$$\mathrm{Mean}(X)=\mu +\sigma {\mu }_1^{\prime }=\mu +\frac{\sigma }{1-c}\text{if }c<1$$

Parametric Variance

$$\mathrm{Variance}(X)={\sigma }^2({\mu }_2^{\prime }-{\mu }_1^{\prime 2})=\frac{{\sigma }^2}{(1-c{)}^2(1-2c)}\text{if }c<1/2$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{2(1+c)\sqrt{1-2c}}{(1-3c)}\text{if }c<1/3$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{3(1-2c)(2c^2+c+3)}{(1-3c)(1-4c)}\text{if }c<1/4$$

Parametric Median

$$\mathrm{Median}(X)=\mu$$

Parametric Mode

$$\mathrm{Mode}(X)=\mu +\frac{\sigma (2^c-1)}{c}$$

Additional Information and Definitions

• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $u:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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