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DISTRIBUTIONS / CONTINUOUS / INVERSE GAUSSIAN 3P

Inverse Gaussian 3P distribution

Portrait

The three-parameter Inverse Gaussian adds a location threshold to a first-passage time. After loc, mu and lambda retain the drift and dispersion interpretation of Inverse Gaussian.

Historical trail

The 3P suffix is a common reliability parameterization extension. Its generative history remains Schrödinger’s and Wald’s first-passage work.

Two useful connections

  • With loc=0 it becomes Inverse Gaussian. It competes with Gamma, Weibull, and Birnbaum–Saunders; its interpretive advantage appears when a first-passage story is plausible.
  • A constant plus a first-passage time is no longer a first passage from the origin, even though the post-threshold shape is retained.

A use case

It can separate fixed assembly or transport time from stochastic time to a threshold. In a sequential process, this is stronger than using loc merely to correct skewness.

Diagnostic advice

Check whether the data contain a real delay or left censoring; both can create an apparent minimum.

A delay before the first-passage clock begins

The third parameter shifts the entire clock. It may represent preparation, instrument latency, or a minimum time preceding the random process. After subtracting loc, the remaining part retains the Inverse Gaussian shape.

That shift is not the same as changing the barrier or drift of Brownian motion, nor is it noncentrality. It is a constant added after generating the first-passage time. The distinction matters when reporting results and extrapolating near the lower endpoint. Because the smallest observation strongly influences loc, small samples or rounded times can make the three-parameter fit unstable. Comparing it with the unshifted version helps expose unnecessary complexity.

Decision guide

A good candidate when: the first-passage process begins after a physical delay or shift.

Compare it with: unshifted Inverse Gaussian and reaction-time models with separate components. Do not automatically attribute every observed minimum latency to the threshold.

References

  • SciPy reference: scipy.stats.invgauss — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Schrödinger, E. (1915). Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung. Physikalische Zeitschrift, 16, 289–295.
  • Wald, A. (1947). Sequential Analysis. Wiley.

Inverse Gaussian 3P Distribution: equations and calculator

Distribution defintion

X∼InverseGaussian3P(μ,λ,Loc)X\sim\mathrm{InverseGaussian_{3P}}\left(\mu,\lambda,\text{Loc}\right)

Distribution domain

x∈(0,∞)x\in\left(0,\infty\right)

Parameters domain and parameters constraints

μ∈R+,λ∈R+,Loc∈R\mu\in\mathbb{R}^{+},\lambda\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Cumulative distribution function

FX(x)=Φ(λx−Loc(x−Locμ−1))+exp⁡(2λμ)Φ(−λx−Loc(x−Locμ+1))F_{X}\left(x\right)=\Phi\left(\sqrt{\frac{\lambda}{x-\text{Loc}}}\left(\frac{x-\text{Loc}}{\mu}-1\right)\right)+\exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x-\text{Loc}}}\left(\frac{x-\text{Loc}}{\mu}+1\right)\right)

Probability density function

fX(x)=λ2π(x−Loc)3exp⁡[−λ(x−μ−Loc)22μ2(x−Loc)]f_{X}\left(x\right)=\sqrt\frac{\lambda}{2 \pi (x-\text{Loc})^3} \exp\left[-\frac{\lambda (x-\mu-\text{Loc})^2}{2 \mu^2 (x-\text{Loc})}\right]

Percent point function/Sample

SampleX={Loc+x0ifu2⩽μμ+x0Loc+μ2x0ifu2⩾μμ+x0\text{Sample}_{X}=\left\{\begin{array}{cl} \text{Loc}+x_{0} \quad \text{if} \quad u_{2}\leqslant\frac{\mu}{\mu+x_{0}}\\ \text{Loc}+\frac{\mu^{2}}{x_{0}} \quad \text{if} \quad u_{2}\geqslant \frac{\mu}{\mu+x_{0}} \end{array} \right.

Non-central parametric moments

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

Parametric mean

Mean(X)=μ1′=Loc+μ\mathrm{Mean}(X)=\mu'_{1}=\text{Loc}+\mu

Parametric variance

Variance(X)=μ2′−μ1′2=μ3λ\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\mu^3}{\lambda}

Parametric skewness

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

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+15μλ\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}}=3+\frac{15 \mu}{\lambda}

Parametric median

Median(X)=FX−1(12)\mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right)

Parametric mode

Mode(X)=Loc+μ[(1+9μ24λ2)12−3μ2λ]\mathrm{Mode}(X)=\text{Loc}+\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
Loc:Location parameter\text{Loc}:\text{Location parameter}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
x0=μ+μ2[Φ−1(u1)]22λ−μ2λ4μλ[Φ−1(u1)]2+μ2([Φ−1(u1)]2)2x_{0}=\mu+\frac{\mu^2 [\Phi^{-1}\left(u_{1}\right)]^{2}}{2\lambda}-\frac{\mu}{2\lambda}\sqrt{4\mu \lambda [\Phi^{-1}\left(u_{1}\right)]^{2}+\mu^2 ([\Phi^{-1}\left(u_{1}\right)]^{2})^2}
u1:Uniform[0,1] random varibleu_{1}:\text{Uniform[0,1] random varible}
u2:Uniform[0,1] random varibleu_{2}:\text{Uniform[0,1] random varible}