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Weibull 3P distribution

Why it exists

The three-parameter Weibull adds a location threshold to Weibull lifetimes. It distinguishes a minimum time before failure hazard begins to act.

Alternative names

It is also called three-parameter Weibull, shifted Weibull, or Weibull with minimum life.

Constructions and limits

At loc=0 it becomes two-parameter Weibull. It competes with Gamma 3P and Exponential 2P for latent lifetimes; shape retains its ageing interpretation.

Applications

A physical loc can represent time before a part accumulates damage. In strength data it may be a theoretical lower limit below which fracture cannot occur; both require process evidence.

Location is notoriously difficult to identify in small samples and can produce very different fits with the same central mass.

The tempting failure threshold

Adding loc allows the claim that no failure occurs before a particular time or strength level. That claim may be physical, such as a load too small to break a part, or administrative, such as preparation time before exposure begins.

The threshold is tempting because it often moves close to the smallest observation and improves the visible fit. It can also absorb rounding, sample selection, or a poorly described left tail. Maximum likelihood estimation has known difficulties, and uncertainty about loc can be highly asymmetric. Responsible analysis compares two and three parameters, examines the likelihood profile, and avoids treating the observed minimum as a known natural bound.

Decision guide

A good candidate when: Weibull lifetimes begin after a physical operating or strength threshold.

Compare it with: unshifted Weibull. The 3P Weibull is sensitive to the earliest failure; confirm its threshold using engineering knowledge and likelihood profiles before extrapolating reliability.

References

  • SciPy reference: scipy.stats.weibull_min — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18, 293–297.
  • Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed. Wiley.

Weibull 3P Distribution: equations and calculator

Distribution defintion

X∼Weibull3P(α,Loc,β)X\sim\mathrm{Weibull_{3P}}\left(\alpha,\text{Loc},\beta\right)

Distribution domain

x∈[Loc,∞)x\in [\text{Loc},\infty)

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−e−z(x)αF_{X}\left(x\right)=1-e^{-z(x)^\alpha}

Probability density function

fX(x)=αβz(x)α−1e−z(x)αf_{X}\left(x\right)=\frac{\alpha}{\beta}z(x)^{\alpha-1}e^{-z(x)^\alpha}

Percent point function/Sample

FX−1(u)=Loc+β(−ln⁡(1−u))1/αF^{-1}_{X}\left(u\right)=\text{Loc}+\beta(-\ln(1-u))^{1/\alpha}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=βαΓ(1+kα)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\beta^\alpha \Gamma\left(1+\frac{k}{\alpha}\right)

Parametric mean

Mean(X)=Loc+μ~1′=Loc+β Γ(1+1/α)\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}=\text{Loc}+\beta \ \Gamma(1+1/\alpha)

Parametric variance

Variance(X)=μ~2′−μ~1′2=β2[Γ(1+2/α)−(Γ(1+1/α))2]\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}=\beta^2\left[\Gamma\left(1+2/\alpha\right)-\left(\Gamma\left(1+1/\alpha\right)\right)^2\right]

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\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}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\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}}

Parametric median

Median(X)=Loc+β(ln⁡(2))1/α\mathrm{Median}(X)=\text{Loc}+\beta(\ln(2))^{1/\alpha}

Parametric mode

Mode(X)=Loc+{β(α−1α)1/αif α>10if α≤1\mathrm{Mode}(X)=\text{Loc}+\left\{\begin{array}{cl} \beta\left(\frac{\alpha-1}{\alpha}\right)^{1/\alpha} & \text{if }\alpha>1\\ 0 & \text{if } \alpha\leq 1 \end{array} \right.

Additional information and definitions

X~∼Weibull(α,β)\tilde{X}\sim\mathrm{Weibull}\left(\alpha,\beta\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
β:Scale parameter\beta:\text{Scale parameter}
z(x)=(x−Loc)/βz\left(x\right)=\left(x-\text{Loc}\right)/\beta
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}