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Gamma 3P distribution

The idea in one sentence

The three-parameter Gamma adds a location threshold to Gamma. After loc, the variable keeps a positive, skewed shape that can be read as baseline time plus accumulated variability.

Support and interpretation

The support starts at loc. alpha sets shape and beta scale; the 3P suffix adds an origin rather than a new special function.

Its place in the family

With loc=0 it is two-parameter Gamma. Integer alpha links it to shifted Erlang; it can also compete with Weibull and lognormal for positive times.

An applied reading

A meaningful loc can represent time before the random mechanism begins. If it merely fixes skewness, compare two-parameter Gamma and Weibull before interpreting the boundary.

The sample minimum is not necessarily loc; with finite data the threshold estimate can be highly sensitive to one observation.

Random accumulation beyond a baseline amount

Gamma 3P writes an observation as a fixed baseline plus a positive Gamma amount. In hydrology this might be base flow accompanied by random input; in reliability it might be a minimum time followed by accumulated damage.

This reading differs from a Gamma variable truncated by an instrument. A shift says the entire population begins at loc; truncation says values exist below a boundary but cannot be observed. Since Gamma already changes shape substantially through alpha, adding a threshold can create parameter tradeoffs. Stable fitting needs information near the endpoint and an external reason to believe in it. Otherwise an ordinary or Generalized Gamma may be more parsimonious.

Decision guide

A good candidate when: Gamma accumulation starts after a physical origin that is unknown but estimable.

Compare it with: Gamma with an origin fixed from domain knowledge. A free location often correlates with shape and can destabilize extrapolation near the threshold.

References

Gamma 3P Distribution: equations and calculator

Distribution defintion

X∼Gamma3P(α,Loc,β)X\sim\mathrm{Gamma_{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)=P(α,x−Locβ)=1Γ(α)γ(α,x−Locβ)F_{X}\left(x\right)=\text{P}\left(\alpha,\frac{x-\text{Loc}}{\beta}\right)=\frac{1}{\Gamma(\alpha)} \gamma\left(\alpha,\frac{x-\text{Loc}}{\beta}\right)

Probability density function

fX(x)=1Γ(α)βα(x−Loc)α−1e−x−Locβf_{X}\left(x\right)=\frac{1}{\Gamma(\alpha) \beta^\alpha} (x-\text{Loc})^{\alpha-1} e^{-\frac{x-\text{Loc}}{\beta}}

Percent point function/Sample

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

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=βkΓ(k+α)Γ(α)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\beta^k\frac{\Gamma(k+\alpha)}{\Gamma(\alpha)}

Parametric mean

Mean(X)=Loc+μ~1′=Loc+αβ\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}=\text{Loc}+\alpha \beta

Parametric variance

Variance(X)=μ~2′−μ~1′2=αβ2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}=\alpha \beta^2

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2α\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=\frac{2}{\sqrt{\alpha}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+6α\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=3+\frac{6}{\alpha}

Parametric median

Median(X)=Loc+(α−1)βif α>1\mathrm{Median}(X)=\text{Loc}+(\alpha-1)\beta \quad \text{if }\alpha>1

Parametric mode

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

Additional information and definitions

X~∼Gamma(α,β)\tilde{X}\sim\mathrm{Gamma}\left(\alpha,\beta\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}