Log-Logistic Distribution#

Univariate, Continuous, Asymmetric, Non-Negative, Extreme values, Heavy-tailed

The log-logistic distribution, also known as the Fisk distribution, is a continuous probability distribution that is used to model non-negative random variables. It is characterized by two parameters: a scale parameter (\(\alpha\)), which is also the median of the distribution, and a shape parameter (\(\beta\)). The dispersion of the distribution decreases as \(\beta\) increases.

The log-logistic distribution is often used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment. It has been used in hydrology for modelling stream flow rates and precipitation. It is also used in reliability engineering to model the lifetime of components and systems.

Key properties and parameters#

Support

\(x \in [0, \infty)\)

Mean

\({\alpha\,\pi/\beta \over \sin(\pi/\beta)}\) if \(\beta>1\), else undefined

Variance

\(\alpha^2 \left(2b / \sin 2b -b^2 / \sin^2 b \right), \quad \beta>2\)

Parameters:

  • \(\alpha\) : (float) Scale parameter, \(\alpha > 0\).

  • \(\beta\) : (float) Shape parameter, \(\beta > 0\).

Probability Density Function (PDF)#

\[ f(x|\alpha, \beta) = \frac{ (\beta/\alpha)(x/\alpha)^{\beta-1}}{\left( 1+(x/\alpha)^{\beta} \right)^2} \]
/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/c/cmodule.py:2986: UserWarning: PyTensor could not link to a BLAS installation. Operations that might benefit from BLAS will be severely degraded.
This usually happens when PyTensor is installed via pip. We recommend it be installed via conda/mamba/pixi instead.
Alternatively, you can use an experimental backend such as Numba or JAX that perform their own BLAS optimizations, by setting `pytensor.config.mode == 'NUMBA'` or passing `mode='NUMBA'` when compiling a PyTensor function.
For more options and details see https://pytensor.readthedocs.io/en/latest/troubleshooting.html#how-do-i-configure-test-my-blas-library
  warnings.warn(
Log-Logistic Distribution PDF

Cumulative Distribution Function (CDF)#

\[ F(x|\alpha, \beta) = \frac{1}{1 + (x/\alpha)^{-\beta}} \]
Log-Logistic Distribution CDF

See also

Common Alternatives:

  • Log-Normal Distribution - The log-logistic distribution is similar to the log-normal distribution, but with heavier tails.

Related Distributions:

  • Logistic Distribution - If a random variable \(X\) is distributed as a log-logistic distribution, then its logarithm \(\log(X)\) follows a logistic distribution.

References#