Surf Wiki
Save to docs
general/continuous-distributions

From Surf Wiki (app.surf) — the open knowledge base

Shifted log-logistic distribution

Probability distribution

Probability distribution

name =Shifted log-logistic | type =density \mu \in (-\infty,+\infty) , location (real)

\sigma \in (0,+\infty) , scale (real)

\xi\in (-\infty,+\infty) , shape (real)| support = x \geqslant \mu -\sigma/\xi,;(\xi 0)

x \leqslant \mu -\sigma/\xi,;(\xi

x \in (-\infty, +\infty) ,;(\xi = 0)| pdf =\frac{(1+\xi z)^{-(1/\xi +1)}}{\sigma\left(1 + (1+\xi z)^{-1/\xi}\right)^2}

where z=(x-\mu)/\sigma,| cdf =\left(1+(1 + \xi z)^{-1/\xi}\right)^{-1} ,

where z=(x-\mu)/\sigma,| mean =\mu + \frac{\sigma}{\xi}(\alpha \csc(\alpha)-1)

where \alpha= \pi \xi, | median =\mu ,| mode =\mu + \frac{\sigma}{\xi}\left[\left(\frac{1-\xi}{1+\xi}\right)^\xi - 1 \right] | variance = \frac{\sigma^2}{\xi^2}[2\alpha \csc(2 \alpha) - (\alpha \csc(\alpha))^2]

where \alpha= \pi \xi, | skewness =| kurtosis =| entropy =| mgf =| char =|

The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. It has also been called the generalized logistic distribution,{{Citation|title=Regional Frequency Analysis: An Approach Based on L-Moments| first1=Jonathan R. M. | last1=Hosking| first2= James R| last2=Wallis| year=1997| publisher=Cambridge University Press| isbn=0-521-43045-3}} but this conflicts with other uses of the term: see generalized logistic distribution.

Definition

The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter \delta. Thus if X has a log-logistic distribution then X+\delta has a shifted log-logistic distribution. So Y has a shifted log-logistic distribution if \log(Y-\delta) has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) log-logistic.

The properties of this distribution are straightforward to derive from those of the log-logistic distribution. However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distribution, gives more interpretable parameters and also aids their estimation.

In this parameterisation, the cumulative distribution function (CDF) of the shifted log-logistic distribution is : F(x; \mu,\sigma,\xi) = \frac{1}{ 1 + \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}} for 1 + \xi(x-\mu)/\sigma \geqslant 0, where \mu\in\mathbb R is the location parameter, \sigma0, the scale parameter and \xi\in\mathbb R the shape parameter. Note that some references use \kappa = - \xi,! to parameterise the shape.

The probability density function (PDF) is : f(x; \mu,\sigma,\xi) = \frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi +1)}} {\sigma\left[1 + \left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right]^2}, again, for 1 + \xi(x-\mu)/\sigma \geqslant 0.

The shape parameter \xi is often restricted to lie in [-1,1], when the probability density function is bounded. When |\xi|1, it has an asymptote at x = \mu - \sigma/\xi. Reversing the sign of \xi reflects the pdf and the cdf about x=\mu..

  • When \mu = \sigma/\xi, the shifted log-logistic reduces to the log-logistic distribution.
  • When \xi → 0, the shifted log-logistic reduces to the logistic distribution.
  • The shifted log-logistic with shape parameter \xi=1 is the same as the generalized Pareto distribution with shape parameter \xi=1.

Applications

The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency.

Alternate parameterization

An alternate parameterization with simpler expressions for the PDF and CDF is as follows. For the shape parameter \alpha, scale parameter \beta and location parameter \gamma, the PDF is given by

f(x) = \frac{\alpha}{\beta} \bigg(\frac{x-\gamma}{\beta} \bigg) ^{\alpha -1}\bigg(1+\bigg(\frac{x-\gamma}{\beta}\bigg)^\alpha\bigg)^{-2}

The CDF is given by

F(x) = \bigg(1+\bigg(\frac{\beta}{x-\gamma}\bigg)^\alpha\bigg)^{-1}

The mean is \beta \theta \csc(\theta) + \gamma and the variance is \beta^2\theta[2\csc(2\theta)-\theta \csc^2(\theta)], where \theta = \frac{\pi}{\alpha}.

References

References

  1. "EasyFit - Log-Logistic Distribution".
  2. "Guide to Using - RISK7_EN.pdf".
Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

continuous-distributions
Want to explore this topic further?

Ask Mako anything about Shifted log-logistic distribution — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report