PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / LOGLOGISTIC

Loglogistic distribution

The idea in one sentence

The Loglogistic distribution is the exponential of a logistic variable. It produces positive lifetimes or sizes with a power-law tail and a simple CDF, making it important in survival analysis.

Support and interpretation

Its support is the entire positive line. alpha acts as scale and beta controls tail shape; mean and variance require conditions on beta.

Its place in the family

It is a generalized beta type II case and is related to Burr XII, Dagum, and lognormal. Unlike lognormal, its tail is a power law and moments may be undefined.

An applied reading

It is useful when risk rises and then falls, a pattern found in some survival curves. In income work, Fisk is a common name and supports comparison with Pareto in the upper tail.

Check moment existence and do not read alpha as the lognormal parameter: the tails have different mechanisms.

A straight line hidden in survival odds

If log time is Logistic, time is Loglogistic. This construction creates a linear relationship between log cumulative odds and log time. It also provides an elementary CDF and quantile function, a practical advantage in accelerated failure-time models.

Its hazard can rise and then fall, unlike Weibull whose hazard is monotone. Such an arc can be plausible when progressively more resistant units remain alive. The tail follows a power law and not every moment exists: shape must exceed one for a mean and two for a variance. A good fit in the centre can therefore conceal highly unstable estimates of average cost or extreme duration.

Decision guide

A good candidate when: positive times require closed-form quantiles and a hazard that may rise and then fall.

Compare it with: Weibull for monotone hazard and Burr XII for greater tail flexibility. In survival analysis, include censoring when comparing models.

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.
  • Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley.

Loglogistic Distribution: equations and calculator

Distribution defintion

X∼LogLogistic(α,β)X\sim\mathrm{LogLogistic}\left(\alpha,\beta\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

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

Cumulative distribution function

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

Probability density function

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

Percent point function/Sample

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

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=αkBeta(1−k/β,1+k/β)=αk kπ/βsin⁡(kπ/β)\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{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)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\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)=α\mathrm{Median}(X)=\alpha

Parametric mode

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

Additional information and definitions

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}