{{Short description|Continuous probability distribution for a non-negative random variable}} {{Probability distribution | name =Log-logistic| | type =density| | pdf_image =Loglogisticpdf.svg | pdf_caption =<math>\alpha=1,</math> values of <math>\beta</math> as shown in legend | cdf_image =Loglogisticcdf.svg | cdf_caption = <math>\alpha=1,</math> values of <math>\beta</math> as shown in legend | parameters =<math>\alpha>0</math> [[scale parameter|scale]]<br /><math>\beta> 0</math> [[shape parameter|shape]] | support =<math>x\in[0,\infty)</math> | pdf =<math> \frac{ (\beta/\alpha)(x/\alpha)^{\beta-1} } { \left (1+(x/\alpha)^{\beta} \right)^2 }</math> | cdf =<math>{ 1 \over 1+(x/\alpha)^{-\beta} }</math> | quantile =<math>\alpha\left( \frac{p}{1-p} \right)^{1/\beta}.</math> | mean =<math>{\alpha\,\pi/\beta \over \sin(\pi/\beta)}</math><br />if <math>\beta>1</math>, else undefined | median =<math>\alpha\,</math> | mode =<math>\alpha\left(\frac{\beta-1}{\beta+1}\right)^{1/\beta}</math><br />if <math>\beta> 1</math>, 0 otherwise | variance =[[#Moments|See main text]] | entropy =<math>\ln \alpha\ - \ln \beta\ + 2</math> | mgf = <math>\beta\alpha^{-\beta}\int_{0}^{\infty}\frac{e^{tx}x^{\beta-1}}{(1+(x/\alpha)^{\beta})^{2}}dx</math><ref name=mgf_cite>{{cite web|url=http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Loglogistic.pdf|title=Log-Logistic distribution|first=Larry|last=Leemis|publisher=College of William & Mary}}</ref><math>= \sum_{n=0}^{\infty}\frac{(\alpha t)^{n}}{n!}\Beta(1+\frac{n}{\beta},1-\frac{n}{\beta})</math> where <math>\Beta</math> is the [[Beta function]].<ref name=mgf_cite2>{{cite journal|last1=Ekawati|first1=D.|last2=Warsono|last3=Kurniasari|first3=D.|year=2014|title=On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution|journal=IPTEK, the Journal for Technology and Science|volume=25|issue=3|pages=78–82}}</ref> | char = <math>\beta\alpha^{-\beta}\int_{0}^{\infty}\frac{e^{itx}x^{\beta-1}}{(1+(x/\alpha)^{\beta})^{2}}dx</math><ref name=mgf_cite /><math>= \sum_{n=0}^{\infty}\frac{(i\alpha t)^{n}}{n!}\Beta(1+\frac{n}{\beta},1-\frac{n}{\beta})</math> where <math>\Beta</math> is the [[Beta function]].<ref name=mgf_cite2 /> | ES = <math>\frac{\alpha}{1-p}\left( \frac{\pi}{\beta} csc\left( \frac{\pi}{\beta} \right) - B_p \left( \frac{1}{\beta}+1, 1 - \frac{1}{\beta} \right) \right)</math> <br /> where <math>B_y(A_1,A_2)=\int _{0}^{y}p^{A_1-1}(1-p)^{A_2-1}dp</math> is the incomplete beta function.<ref name="norton">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer |doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |s2cid=254231768 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27 |archive-date=2023-03-31 |archive-url=https://web.archive.org/web/20230331230821/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |url-status=dead }}</ref> }} In [[probability]] and [[statistics]], the '''log-logistic distribution''' (known as the '''Fisk distribution''' in [[economics]]) is a [[continuous probability distribution]] for a non-negative [[random variable]]. It is 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 also been used in [[hydrology]] to model stream flow and [[precipitation]], in [[economics]] as a simple model of the [[distribution of wealth]] or [[income distribution|income]], and in [[Computer network|networking]] to model the transmission times of data considering both the network and the software.

The log-logistic distribution is the probability distribution of a [[random variable]] whose [[logarithm]] has a [[logistic distribution|logistic]] distribution.↵ It is similar in shape to the [[log-normal distribution]] but has [[Heavy-tailed distribution|heavier tails]] . Unlike the log-normal, its [[cumulative distribution function]] can be written in [[closed form expression|closed form]].

==Characterization== There are several different parameterizations of the distribution in use. The one shown here gives reasonably interpretable parameters and a simple form for the [[cumulative distribution function|cumulative distribution]] function [1].<ref name=Shoukri> {{Citation| title=Sampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation Data| first1=M.M.|last1= Shoukri| first2= I.U.M.| last2=Mian| first3=D.S.| last3=Tracy |journal=[[The Canadian Journal of Statistics]] |volume=16 | year=1988| pages=223–236| issue=3| doi=10.2307/3314729| jstor=3314729}}</ref><ref name=Ashkar06> {{Citation|first1=Fahim |last1=Ashkar|first2= Smail|last2= Mahdi |title=Fitting the log-logistic distribution by generalized moments|journal=[[Journal of Hydrology]]| volume=328| year=2006| pages=694–703| doi=10.1016/j.jhydrol.2006.01.014|issue=3–4|bibcode=2006JHyd..328..694A}}</ref> The parameter <math>\alpha>0</math> is a [[scale parameter]] and is also the [[median]] of the distribution. The parameter <math>\beta>0</math> is a [[shape parameter]]. The distribution is [[unimodal]] when <math>\beta>1</math> its [[Statistical dispersion|dispersion]] decreases as it <math>\beta</math> increases.

The [[cumulative distribution function]] is :<math>\begin{align} F(x; \alpha, \beta) & = { 1 \over 1+(x/\alpha)^{-\beta} } \\[5pt] & = {(x/\alpha)^\beta \over 1+(x/\alpha)^ \beta } \\[5pt] & = {x^\beta \over \alpha^\beta+x^\beta} \end{align}</math> where <math>x>0</math>, <math>\alpha>0</math>, <math>\beta>0.</math>

The [[probability density function]] is :<math>f(x; \alpha, \beta) = \frac{ (\beta/\alpha)(x/\alpha)^{\beta-1} } { \left( 1+(x/\alpha)^{\beta} \right)^2 }</math>

===Alternative parameterization=== An alternative parametrization is given by the pair <math>\mu, s</math> in analogy with the logistic distribution:

: <math>\mu = \ln (\alpha)</math> : <math>s = 1 / \beta</math>

==Properties==

===Moments===

The <math>k</math>throw [[moment (mathematics)|moment]] exists only when <math>k<\beta,</math> when it is given by [3][4].<ref name=TadiBurr80>{{Citation|title=A Look at the Burr and Related Distributions| first=Pandu R.| last=Tadikamalla| journal=International Statistical Review| volume=48| year=1980| pages=337–344|issue=3|doi=10.2307/1402945| jstor=1402945}}</ref> :<math>\begin{align} \operatorname{E}(X^k) & = \alpha^k\operatorname{B}(1-k/\beta, 1+k/\beta) \\[5pt] & = \alpha^k\, {k\pi/\beta \over \sin(k\pi/\beta)} \end{align}</math> where B is the [[beta function|beta]] function. Expressions for the [[expected value|mean]], [[variance]], [[skewness]], and [[kurtosis]] can be derived from this. Writing <math>b=\pi/\beta</math> for convenience, the mean is :<math> \operatorname{E}(X) = \alpha b / \sin b , \quad \beta>1,</math> and the variance is :<math> \operatorname{Var}(X) = \alpha^2 \left( 2b / \sin 2b -b^2 / \sin^2 b \right), \quad \beta>2.</math> Explicit expressions for the skewness and kurtosis are lengthy.<ref> {{Citation|title=A Compendium of Common Probability Distributions|page=A–37|first=Michael P. |last=McLaughlin|url=http://www.causascientia.org/math_stat/Dists/Compendium.pdf|access-date=2008-02-15| year=2001}}</ref> As <math>\beta</math> tends to infinity, the mean tends to the variance and skewness tend to zero, and the excess kurtosis tends to 6/5 (see also [[#Related distributions|related distributions]] below).

===Quantiles===

The [[quantile function]] (inverse cumulative distribution function) is :<math>F^{-1}(p;\alpha, \beta) = \alpha\left( \frac{p}{1-p} \right)^{1/\beta}.</math> It follows that the [[median]] is, the lower [[quartile]] is, and the upper quartile is <math>3^{1/\beta} \alpha</math>.

==Applications==

[[Image:Loglogistichaz.svg|thumb|right|[[Hazard function]]. <math>\alpha=1,</math> values of <math>\beta</math> as shown in legend]]

===Survival analysis===

The log-logistic distribution provides one [[parametric model]] for [[survival analysis]]. Unlike the more commonly used [[Weibull distribution]], it can have a non-[[monotonic]] [[hazard function]]: when <math>\beta>1,</math> the hazard function is [[unimodal]] (when <math>\beta</math>&nbsp;≤&nbsp;1, the hazard decreases monotonically). The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring.<ref> {{Citation | title=Log-Logistic Regression Models for Survival Data | first=Steve |last=Bennett | volume=32 | year=1983 |pages=165–171 | issue=2 | doi=10.2307/2347295 | journal=Journal of the Royal Statistical Society, Series C | jstor=2347295}} </ref> The log-logistic distribution can be used as the basis of an [[accelerated failure time model]] by allowing it <math>\alpha</math> to differ between groups, or more generally by introducing covariates that affect it <math>\alpha</math> but not <math>\beta</math> by modelling it<math>\log(\alpha)</math> as a linear function of the covariates.<ref> {{Citation | title =Modelling Survival Data in Medical Research|first=Dave |last=Collett | year=2003 | edition=2nd | publisher=CRC press| isbn=978-1-58488-325-8}}</ref>

The [[survival function]] is :<math>S(t) = 1 - F(t) = [1+(t/\alpha)^{\beta}]^{-1},\, </math> and so the [[hazard function]] is :<math> h(t) = \frac{f(t)}{S(t)} = \frac{(\beta/\alpha)(t/\alpha)^{\beta-1}} {1+(t/\alpha)^\beta}.</math>

The log-logistic distribution with shape parameter <math>\beta = 1</math> is the marginal distribution of the inter-times in a geometric-distributed [[counting process|counting]] process [8].<ref> {{Citation | title=Some results and applications of geometric counting processes | first1=Antonio |last1=Di Crescenzo | first2=Franco |last2=Pellerey | volume=21 | year=2019 |pages=203–233 | issue=1 | doi=10.1007/s11009-018-9649-9 | journal=Methodology and Computing in Applied Probability| s2cid=254793416 }} </ref>

===Hydrology=== [[File:FitLog-logisticdistr.tif|thumb|Fitted cumulative log-logistic distribution to maximum one-day October rainfalls ]] The log-logistic distribution has been used in hydrology for modeling stream flow rates and precipitation.<ref name=Shoukri/><ref name=Ashkar06/>

Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a [[log-normal distribution|log-normal]] distribution [9]. The log-normal distribution, however, needs a numeric approximation. As the log-logistic distribution, which can be solved analytically, is similar to the log-normal distribution, it can be used instead.

The blue picture illustrates an example of fitting the log-logistic distribution to ranked maximum one-day October rainfalls, and it shows the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by the [[plotting position]] ''r''/(''n''+1) as part of the [[cumulative frequency analysis]] .

===Economics===

The log-logistic has been used as a simple model of the [[distribution of wealth]] or [[income distribution|income]] in [[economics]], where it is known as the Fisk distribution.<ref> {{Citation|last=Fisk|first= P.R.| year=1961| title=The Graduation of Income Distributions |journal= Econometrica| volume=29 |pages=171–185|doi=10.2307/1909287|issue=2|jstor=1909287}}</ref> Its [[Gini coefficient]] is [11].<ref name=KK> {{Citation|last1=Kleiber |first1=C. |last2=Kotz| first2= S| year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences | publisher=Wiley | isbn=978-0-471-15064-0}}</ref>

{{Collapse top|title=Derivation of Gini coefficient}} The Gini coefficient for a continuous probability distribution takes the form:

:<math>G = {1\over{\mu}}\int_{0}^{\infty}F(1-F)dx</math>

where <math>F</math> is the CDF of the distribution and <math>\mu</math> is the expected value. For the log-logistic distribution, the formula for the Gini coefficient becomes:

:<math>G = {\sin(\pi/\beta) \over{\alpha\pi/\beta}} \int_{0}^{\infty}{dx\over{[1+(x/\alpha)^{-\beta}][1+(x/\alpha)^{\beta}]}}</math>

Defining the substitution <math>z = x/\alpha</math> leads to the simpler equation:

:<math>G = {\sin(\pi/\beta) \over{\pi/\beta}} \int_{0}^{\infty}{dz\over{(1+z^{-\beta})(1+z^{\beta})}}</math>

And making the substitution <math>u = 1/(1 + z^{\beta})</math> further simplifies the Gini coefficient formula to:

:<math>G = {\sin(\pi/\beta)\over{\pi}} \int_{0}^{1}u^{-1/\beta}(1-u)^{1/\beta}du</math>

The integral component is equivalent to the standard [[beta function]] <math>\text{B}(1-1/\beta,1+1/\beta)</math>. The beta function may also be written as:

:<math>\text{B}(x,y) = {\Gamma(x)\Gamma(y)\over{\Gamma(x+y)}}</math>

where <math>\Gamma(\cdot)</math> is the [[gamma function]]. Using the properties of the gamma function, it can be shown that:

:<math>\text{B}(1-1/\beta,1+1/\beta) = {1\over{\beta}}\Gamma(1-1/\beta)\Gamma(1/\beta)</math>

From [[Euler's reflection formula]], the expression can be simplified further:

:<math>\text{B}(1-1/\beta,1+1/\beta) = {1\over{\beta}} {\pi\over{\sin(\pi/\beta)}}</math>

Finally, we may conclude that the Gini coefficient for the log-logistic distribution <math>G = 1/\beta</math>. {{Collapse bottom}}

===Networking===

The log-logistic has been used as a model for the period of time beginning when some data leaves a software user application in a computer and the response is received by the same application after traveling through and being processed by other computers, applications, and network segments, most or all of them without hard [[Real-time computing|real-time]] guarantees (for example, when an application is displaying data coming from a remote [[sensor]] connected to the Internet). It has been shown to be a more accurate probabilistic model for that than the [[log-normal distribution]] or others, as long as abrupt changes of regime in the sequences of those times are properly detected.<ref>{{Citation|author1=Gago-Benítez, A. |author2=Fernández-Madrigal J.-A., Cruz-Martín, A. |title=Log-Logistic Modeling of Sensory Flow Delays in Networked Telerobots|journal=IEEE Sensors Journal |volume=13 |issue=8 |year=2013|publisher=IEEE Sensors 13(8)|pages=2944–2953 |doi=10.1109/JSEN.2013.2263381 |bibcode=2013ISenJ..13.2944G |s2cid=47511693 }}</ref>

==Related distributions==

* If <math> X \sim \operatorname{LL}(\alpha,\beta)</math> then <math> kX \sim \operatorname{LL}(k \alpha, \beta).</math> * If <math> X \sim \operatorname{LL}(\alpha, \beta)</math> then <math> X^k \sim \operatorname{LL}(\alpha^k, \beta/|k|).</math> * <math> \operatorname{LL}(\alpha,\beta) \sim \textrm{Dagum}(1,\alpha,\beta)</math> ([[Dagum distribution]]). * <math> \operatorname{LL}(\alpha,\beta) \sim \textrm{SinghMaddala}(1,\alpha,\beta)</math> ([[Singh–Maddala distribution]] ). * <math>\textrm{LL}(\gamma,\sigma) \sim \beta'(1,1,\gamma,\sigma)</math> ([[Beta prime distribution]]). *If ''X'' has a log-logistic distribution with scale parameter <math>\alpha</math> and shape parameter, <math>\beta</math> then ''Y''&nbsp;=&nbsp;log(''X'') has a [[logistic distribution]] with location parameter <math>\log(\alpha)</math> and scale parameter <math>1/\beta.</math> *As the shape parameter <math>\beta</math> of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow) [[logistic distribution]]. Informally: ::<math>\operatorname{LL}(\alpha, \beta) \to L(\alpha,\alpha/\beta) \quad \text{as} \quad \beta \to \infty.</math>

*The log-logistic distribution with shape parameter <math>\beta=1</math> and scale parameter <math>\alpha</math> is the same as the [[Pareto distribution#Generalized Pareto distributions|generalized Pareto distribution]] with location parameter <math>\mu=0</math>, shape parameter <math>\xi=1</math> and scale parameter <math>\alpha:</math> ::<math>\operatorname{LL}(\alpha,1) = \operatorname{GPD}(1,\alpha,1).</math>

*The addition of another parameter (a shift parameter) formally results in a [[shifted log-logistic distribution]], but this is usually considered in a different parameterization so that the distribution can be bounded above or bounded below.

===Generalizations=== Several different distributions are sometimes referred to as the '''generalized log-logistic distribution''', as they contain the log-logistic as a special case.<ref name=KK/> These include the [[Burr Type XII distribution]] (also known as the ''Singh–Maddala distribution'') and the [[Dagum distribution]], both of which include a second shape parameter. Both are in turn special cases of the even more general ''generalized beta distribution of the second kind''. Another more straightforward generalization of the log-logistic is the [[shifted log-logistic distribution]] .

Another '''generalized log-logistic distribution''' is the log-transform of the [[metalog distribution]], in which power series expansions in terms of <math>p</math> are substituted for [[logistic distribution]] parameters <math>\mu</math>. The resulting [[Metalog distribution#/Unbounded, semibounded, and bounded metalog distributions|log-metalog distribution]] is highly shape flexible, has a simple closed-form [[probability density function|PDF]] and [[quantile function]], can be fit to data with linear least squares, and subsumes the log-logistic distribution as a special case.

==See also==

*[[List of probability distributions#Supported on semi-infinite intervals, usually .5B0.2C.E2.88.9E.29|Probability distributions: List of important distributions supported on semi-infinite intervals]]

==References== {{reflist}}

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Log-Logistic Distribution}} [[Category:Continuous distributions]] [[Category:Survival analysis]] [[Category:Probability distributions with non-finite variance]] [[Category:Economic inequality]]