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DISTRIBUTIONS / CONTINUOUS / LOGLOGISTIC 3P

Loglogistic 3P distribution

Why it exists

The three-parameter Loglogistic shifts loglogistic to represent lifetimes or sizes beginning after a location. It retains the log-scale logistic core and power-law tail.

Alternative names

It is also called three-parameter log-logistic or shifted Fisk.

Constructions and limits

With loc=0 it becomes loglogistic. It also links to Burr XII and GB2; compared with Weibull, it permits a hazard that rises and then falls.

Applications

It can represent a lifetime that cannot start before a technical delay. In survival studies, distinguish such a threshold from late entry into the study.

A free loc can confuse left censoring with a physical minimum; document how early observations were obtained.

The logistic clock receives a starting time

Loglogistic 3P adds a threshold to a Loglogistic time. After subtracting loc, the logarithm of the excess retains a Logistic shape. This permits a minimum latency together with a hazard that first rises and then falls.

The logarithm only exists for positive excesses, so observations at or very near the threshold dominate the fit. Rounding and instrument resolution can imitate evidence for a real location. A shift also leaves the power-law tail intact and cannot create moments forbidden by the shape. Before reporting means or variances, check their existence conditions and the joint uncertainty in threshold and shape.

Decision guide

A good candidate when: a Loglogistic duration starts only after a physical threshold or minimum latency.

Compare it with: unshifted Loglogistic and 3P Weibull. Validate the threshold from process knowledge rather than only the minimum observation.

References

  • SciPy reference: scipy.stats.fisk — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Fisk, P. R. (1961). The graduation of income distributions. Econometrica, 29(2), 171–185.
  • Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed. Wiley.

Loglogistic 3P Distribution: equations and calculator

Distribution defintion

X∼LogLogistic3P(Loc,α,β)X\sim\mathrm{LogLogistic_{3P}}\left(\text{Loc},\alpha,\beta\right)

Distribution domain

x∈[Loc,∞)x\in [\text{Loc},\infty)

Parameters domain and parameters constraints

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

Cumulative distribution function

FX(x)=11+((x−Loc)/α)−βF_{X}\left(x\right)=\frac{1}{1+((x-\text{Loc})/\alpha)^{-\beta}}

Probability density function

fX(x)=(β/α)((x−Loc)/α)β−1(1+((x−Loc)/α)β)2f_{X}\left(x\right)=\frac{ (\beta/\alpha)((x-\text{Loc})/\alpha)^{\beta-1} }{ \left (1+((x-\text{Loc})/\alpha)^{\beta}\right)^2 }

Percent point function/Sample

FX−1(u)=Loc+α(u1−u)1/βF^{-1}_{X}\left(u\right)=\text{Loc}+\alpha\left(\frac{u}{1-u}\right)^{1/\beta}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=αkBeta(1−k/β,1+k/β)=αk kπ/βsin⁡(kπ/β)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\alpha^k \text{Beta}(1-k/\beta,1+k/\beta)=\alpha^k\,\frac{k\pi/\beta}{\sin(k\pi/\beta)}

Parametric mean

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

Parametric variance

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

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+α\mathrm{Median}(X)=\text{Loc}+\alpha

Parametric mode

Mode(X)=Loc+α(β−1β+1)1/β\mathrm{Mode}(X)=\text{Loc}+\alpha\left(\frac{\beta-1}{\beta+1}\right)^{1/\beta}

Additional information and definitions

X~∼LogLogistic(α,β)\tilde{X}\sim\mathrm{LogLogistic}\left(\alpha,\beta\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
α:Scale parameter\alpha:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}