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DISTRIBUTIONS / CONTINUOUS / INVERSE GAMMA 3P

Inverse Gamma 3P distribution

Quick view

The three-parameter Inverse Gamma adds location to Inverse Gamma. It models a variable beginning at a threshold and then allowing a much longer right tail than an exponential.

If you are coming from another distribution

With loc=0 it becomes two-parameter Inverse Gamma. It remains related to Beta prime and F, but location removes the simple ratio-from-zero interpretation.

History and terminology

The 3P form is common in lifetime software and reliability work; its core is the reciprocal transform of Gamma. It is not a noncentral Inverse Gamma.

A familiar situation

It can represent a duration exposed to risk only after a minimum time and then having a heavy tail. In Bayesian inference, adding loc usually removes standard conjugacy and should be stated explicitly.

Fitting with care

Location and scale can be strongly correlated: compare with the two-parameter model and use profile intervals.

A heavy tail raised above a floor

The 3P version places the entire Inverse Gamma law above loc. It suits a quantity with a known floor whose excess retains a power-law tail. The shift changes the start of the support but not the moment existence conditions for that excess.

Estimating floor, shape, and scale together is delicate. Moving loc near the smallest observation radically changes the implied reciprocals and can create unstable likelihood maxima. A bound imposed by physical calibration supplies more information than one inferred from the sample alone. Without such knowledge, compare the unshifted Inverse Gamma and other heavy-tailed families before extrapolating.

Decision guide

A good candidate when: an Inverse Gamma structure operates above a physically defensible baseline.

Compare it with: unshifted Inverse Gamma. Heavy tails combined with a free loc can be hard to identify; use profiles and sensitivity to extremes.

References

Inverse Gamma 3P Distribution: equations and calculator

Distribution defintion

X∼InverseGamma3P(α,Loc,β)X\sim\mathrm{InverseGamma_{3P}}\left(\alpha,\text{Loc},\beta\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

α∈R+,Loc∈R,β∈R+\alpha\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\beta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−γ(α,β/(x−Loc))Γ(α)=1−P(α,βx−Loc)F_{X}\left(x\right)=1-\frac{\gamma(\alpha,\beta/(x-\text{Loc}))}{\Gamma(\alpha)}=1-\text{P}\left(\alpha,\frac{\beta}{x-\text{Loc}}\right)

Probability density function

fX(x)=βαΓ(α)(x−Loc)−α−1exp⁡(−βx−Loc)f_{X}\left(x\right)=\frac{\beta^\alpha}{\Gamma(\alpha)} (x-\text{Loc})^{-\alpha-1} \exp\left(-\frac{\beta}{x-\text{Loc}}\right)

Percent point function/Sample

FX−1(u)=Loc+βP−1(α,1−u)F^{-1}_{X}\left(u\right)=\text{Loc}+\frac{\beta}{\text{P}^{-1}\left(\alpha,1-u\right)}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=Γ(α−k)Γ(α)=1(α−1)⋯(α−k)if α>k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{\Gamma(\alpha-k)}{\Gamma(\alpha)}=\frac{1}{(\alpha-1) \cdots (\alpha-k)}\quad \text{if } \alpha>k

Parametric mean

Mean(X)=Loc+βμ1′\mathrm{Mean}(X)=\text{Loc}+\beta\mu'_{1}

Parametric variance

Variance(X)=β2(μ2′−μ1′2)\mathrm{Variance}(X)=\beta^{2}(\mu'_{2}-\mu'^{2}_{1})

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)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}}

Parametric median

Median(X)=Loc+βP−1(α,12)\mathrm{Median}(X)=\text{Loc}+\frac{\beta}{\text{P}^{-1}\left(\alpha,\frac{1}{2}\right)}

Parametric mode

Mode(X)=Loc+βα+1\mathrm{Mode}(X)=\text{Loc}+\frac{\beta}{\alpha+1}

Additional information and definitions

X~∼InverseGamma3P(α,0,1)\tilde{X}\sim\mathrm{InverseGamma_{3P}}\left(\alpha,0,1\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
β:Scale parameter\beta:\text{Scale parameter}
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}