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Loggamma distribution

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Loggamma models a real-valued variable whose exponential or related transform has Gamma structure. It can represent skewness that normal cannot, with a shape parameter c.

Feature Reading
Support Its support is the full real line. mu locates and sigma scales in the implemented parameterization; c determines which side is elongated.
Shape It is related to Gamma and lognormal, but is not simply a normal log-variable: the transform and c create a different asymmetry. It can also model errors on a log scale.

Origin and terminology

The family is studied through logarithmic transforms of Gamma and in catalogues of continuous distributions. Its identity depends more on the chosen transform than on one historical episode.

In real models

  • transformed errors and logarithmic measurements
  • durations, sizes, and skewed residuals on the real line

Comparisons

Check conventions for c, location, and scale before interpreting skewness: identical-looking shapes can receive opposite signs in other packages.

Taking logs of Gamma opens the entire line

If a positive variable is Gamma, its logarithm is Loggamma. The transformation opens the support to the whole real line and turns multiplicative scales into shifts. When shape equals one, the law is related through reflection and location to Gumbel because the logarithm of an Exponential variable produces an extreme-value form.

The name can be ambiguous: in survival analysis, log-gamma sometimes denotes other parameterizations or extensions. Phitter uses the explicit logarithmic transformation. Ordinary moments of the logged variable are expressed through polygamma functions and all exist despite the asymmetry. When interpreting original data, remember that exponentiating changes means, intervals, and additive effects.

Decision guide

A good candidate when: the real-valued variable is the logarithm of a Gamma quantity and log-scale skewness is interpretable.

Compare it with: Normal and Gumbel. Confirm orientation and shape convention; “log-gamma” denotes different parameterizations across disciplines.

References

Loggamma Distribution: equations and calculator

Distribution defintion

X∼LogGamma(c,μ,σ)X\sim\mathrm{LogGamma}\left(c,\mu,\sigma\right)

Distribution domain

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

Parameters domain and parameters constraints

c∈R+,μ∈R,σ∈R+c\in\mathbb{R}^{+},\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=γ(c,ex)Γ(c)=P(c,ez(x))F_{X}\left(x\right)=\frac{\gamma\left(c,e^{x}\right)}{\Gamma\left(c\right)}=\text{P}\left(c,e^{z(x)}\right)

Probability density function

fX(x)=exp⁡(cz(x)−ez(x))σΓ(c)f_{X}\left(x\right)=\frac{\exp\left(cz(x)-e^{z(x)}\right)}{\sigma\Gamma\left(c\right)}

Percent point function/Sample

FX−1(u)=μ+σln⁡(P−1(u,c))F^{-1}_{X}\left(u\right)=\mu+\sigma\ln\left(\text{P}^{-1}\left(u,c\right)\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)=μ1′=μ+σψ0\mathrm{Mean}(X)=\mu'_{1}=\mu+\sigma\psi_{0}

Parametric variance

Variance(X)=μ2′−μ1′2=α2ψ1(c)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\alpha^{2}\psi_{1}\left(c\right)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=ψ2(c)ψ1(c)\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)}{\psi_{1}\left(c\right)}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=ψ3(c)ψ1(c)\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{\psi_{3}\left(c\right)}{\psi_{1}\left(c\right)}

Parametric median

Median(X)=μ+σln⁡(P−1(1/2,c))\mathrm{Median}(X)=\mu+\sigma\ln\left(\text{P}^{-1}\left(1/2,c\right)\right)

Parametric mode

Mode(X)=μ+σln⁡(c)\mathrm{Mode}(X)=\mu+\sigma\ln(c)

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}
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}