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Generalized Logistic distribution

What it describes

The Generalized Logistic extends the logistic curve with a shape parameter that allows skewness. It can give the two tails and central concentration different behaviour while retaining a sigmoid CDF.

In this implementation the support is the entire real line. loc provides a location reference, scale sets the unit, and c controls asymmetry.

History and names

The classical logistic originated in Verhulst’s population-growth model. Generalizations appeared in survival and income analysis to allow non-symmetric profiles.

How it connects to other distributions

With a suitable shape value it reduces to ordinary logistic. It is also related to Beta-generated families and flexible-hazard survival models.

Where it appears

  • population growth and asymmetric sigmoid curves
  • survival, income, and cumulative response models with unequal tails

Fitting cautions

Check shape conventions before comparing implementations because the same name denotes different families. Strictly positive data require a justified transformation or an appropriate positive-support model.

A power that breaks Logistic symmetry

In Phitter’s parameterization, the standard Logistic CDF is raised to c. When c is an integer, this is exactly the distribution of the maximum of c independent Logistic variables. That construction explains the shift in mass and resulting asymmetry.

At c=1, ordinary Logistic is recovered. At other values, loc is generally neither mean nor median, so calling it the centre can mislead. The Generalized Logistic name covers several incompatible families in the literature, including growth and extreme-value versions. Before transferring formulas or applications, the CDF is a safer identifier than the name.

Decision guide

A good candidate when: a real-line distribution with a sigmoid CDF and power-controlled asymmetry is required.

Compare it with: Logistic and Gumbel. Check the exact convention: “Generalized Logistic” names several incompatible families, and loc is not necessarily the mean or median.

References

Generalized Logistic Distribution: equations and calculator

Distribution defintion

X∼GeneralizedLogistic(c,Loc,Sc)X\sim\mathrm{GeneralizedLogistic}\left(c,\text{Loc},\text{Sc}\right)

Distribution domain

x∈(−∞,+∞)x\in\left(-\infty,+\infty\right)

Parameters domain and parameters constraints

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

Cumulative distribution function

FX(x)=1(1+exp⁡(−z(x)))cF_{X}\left(x\right)=\frac{1}{\left(1+\exp\left(-z(x)\right)\right)^{c}}

Probability density function

fX(x)=cexp⁡(−z(x))Sc(1+exp⁡(−z(x)))c+1f_{X}\left(x\right)=\frac{c\exp\left(-z(x)\right)}{\text{Sc}\left(1+\exp\left(-z(x)\right)\right)^{c+1}}

Percent point function/Sample

FX−1(u)=Loc−Scln⁡(u−1/c−1)F^{-1}_{X}\left(u\right)=\text{Loc}-\text{Sc}\ln\left(u^{-1/c}-1\right)

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=Loc+Sc⋅μ~1′=Loc+Sc(γ+ψ0(c))\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\left(\gamma+\psi_{0}\left(c\right)\right)

Parametric variance

Variance(X)=Sc2⋅(μ~2′−μ~1′2)=Sc2(π26+ψ1(c))\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\left(\frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\right)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=ψ2(c)+2ζ(3)(π26+ψ1(c))3/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\left(\frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\right)^{3/2}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=(π415+ψ3(c))(π26+ψ1(c))2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\left(\frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\right)^{2}}

Parametric median

Median(X)=Loc−Scln⁡(21/c−1)\mathrm{Median}(X)=\text{Loc}-\text{Sc}\ln\left(2^{1/c}-1\right)

Parametric mode

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

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
γ:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649
ψ0(x):Digamma function\psi_{0}\left(x\right):\text{Digamma function}
ψn(x):Polygamma function of order n∈N\psi_{n}\left(x\right):\text{Polygamma function of order }n\in\mathbb{N}