GENERALIZED LOGISTIC DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.GeneralizedLogistic({"c": *, "loc": *, "scale": *})

💡 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

$$X\sim \mathrm{GeneralizedLogistic}(c,\text{Loc},\text{Sc})$$

Distribution Domain

$$x\in (\text{Loc},\mathrm{\infty })$$

Parameters Domain and Constraints

$$c\in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R},\text{Sc}\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{1}{{(1+\exp (-z(x)))}^c}$$

Probability Density Function

$$f_X\left(x\right)=\frac{c\exp (-z(x))}{\text{Sc}{(1+\exp (-z(x)))}^{c+1}}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}-\text{Sc}\ln (u^{-1/c}-1)$$

Parametric Centered Moments

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

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot {\tilde{\mu }}_1^{\prime }=\text{Loc}+\text{Sc}(\gamma +{\psi }_0\left(c\right))$$

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2\cdot ({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})={\text{Sc}}^2(\frac{{\pi }^2}{6}+{\psi }_1\left(c\right))$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{{\psi }_2\left(c\right)+2\zeta \left(3\right)}{{(\frac{{\pi }^2}{6}+{\psi }_1\left(c\right))}^{3/2}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{(\frac{{\pi }^4}{15}+{\psi }_3\left(c\right))}{{(\frac{{\pi }^2}{6}+{\psi }_1\left(c\right))}^2}$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}-\text{Sc}\ln (2^{1/c}-1)$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\ln \left(c\right)$$

Additional Information and Definitions

• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $\gamma :\text{Euler-Mascheroni constant}=0.5772156649$
• ${\psi }_0\left(x\right):\text{Digamma function}$
• ${\psi }_n\left(x\right):\text{Polygamma function of order }n\in \mathbb{N}$

Spreadsheet Documents

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