View source for Log-logistic distribution

{{Probability distribution|
   name       =Log-logistic|
   type       =density|
   pdf_image  =[[Image:Loglogisticpdf.svg|thumb|275px|<math>\alpha=1,</math> values of <math>\beta</math> as shown in legend]]|
   cdf_image  =[[Image:Loglogisticcdf.svg|thumb|275px|<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>|
   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]]|
   skewness   =|
   kurtosis   =|
   entropy    =|
   mgf        =|
   char       =|
 }}

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, for example mortality from cancer following diagnosis or treatment. It has also been used in [[hydrology]] to model stream flow and precipitation, and in [[economics]] as a simple model of the [[distribution of wealth]] or [[income distribution|income]].

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

==Characterisation==
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]].<ref name=Shoukri>
{{Citation| title=Sampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation Data| first=M.M.|last= Shoukri| first2= I.U.M.| last2=Mian| first3=D.S.| last3=Tracy |journal=The Canadian Journal of Statistics |volume=16 |number=3| year=1988| pages=223-236| url=http://links.jstor.org/sici?sici=0319-5724%28198809%2916%3A3%3C223%3ASPOEOT%3E2.0.CO%3B2-E}}</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}}</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> and its [[dispersion]] decreases as <math>\beta</math> increases. 

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

==Properties==

===Moments===

The <math>k</math>th raw [[moment (mathematics)|moment]] exists only when <math>k<\beta,</math> when it is given by<ref name=Tadi82>
{{Citation| title=Systems of Frequency Curves Generated by Transformations of Logistic Variables |first=Pandu R. |last=Tadikamalla|first2= Norman L. |last2=Johnson| journal=Biometrika| volume=69 |number=2| year=1982| pages= 461-465| url=http://links.jstor.org/sici?sici=0006-3444%28198208%2969%3A2%3C461%3ASOFCGB%3E2.0.CO%3B2-Y}}</ref><ref name=TadiBurr80>{{Citation|title=A Look at the Burr and Related Distributions| first=Pandu R.| last=Tadikamalla| journal=International Statistical Review| volume=48| number=3| year=1980| pages=337-344| url=http://links.jstor.org/sici?sici=0306-7734%28198012%2948%3A3%3C337%3AALATBA%3E2.0.CO%3B2-Z}}</ref>
:<math>\begin{align}
\operatorname{E}(X^k) 
&  = \alpha^k\,\operatorname{B}(1-k/\beta,\, 1+k/\beta) \\
& = \alpha^k\, {k\,\pi/\beta \over \sin(k\,\pi/\beta)}
\end{align}</math> 
where B() is the [[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 kurtosis and variance 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 <math>\alpha</math>, 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 <math>\alpha</math>, 
the lower [[quartile]] is <math>3^{1/\beta} \alpha </math> 
and the upper quartile is <math>3^{-1/\beta} \alpha</math>.

==Applications==

[[Image:Loglogistichaz.svg|thumb|275px|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 (statistics)#life testing|censoring]].<ref>
{{Citation | title=Log-Logistic Regression Models for Survival Data | first=Steve |last=Bennett | journal=Applied Statistics |volume=32 | number=2 | year=1983 |pages=165-171 |url=http://links.jstor.org/sici?sici=0035-9254%281983%2932%3A2%3C165%3ALRMFSD%3E2.0.CO%3B2-F}}
</ref> 
The log-logistic distribution can be used as the basis of an [[accelerated failure time model]] by allowing <math>\beta</math> to differ between groups, or more generally by introducing covariates that affect <math>\beta</math> but not <math>\alpha</math> by modelling <math>\log(\beta)</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=1584883251}}</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)(x/\alpha)^{\beta-1}}
                                       {[1+(x/\alpha)^{\beta}]}.</math>

===Hydrology===

The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation.<ref name=Shoukri/><ref name=Ashkar06/>

===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|url=http://links.jstor.org/sici?sici=0012-9682%28196104%2929%3A2%3C171%3ATGOID%3E2.0.CO%3B2-Y}}</ref>
Its [[Gini coefficient]] is <math>1/\beta</math>.<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=0471150649}}</ref>

==Related distributions==

*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>\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, as <math>\beta</math>→∞, 
:<math>LL(\alpha, \beta) \to 
       L(\alpha,\alpha/\beta). </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 distribution|generalized Pareto distribution]] with location parameter <math>\mu=0</math>, shape parameter <math>\xi=1</math> and scale parameter <math>\alpha:</math>
:<math>LL(\alpha,1) = GPD(1,\alpha,1).\,</math>

===Generalizations===

{{redirects|Generalizations|"Generalized log-logistic distribution"}}

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 given in the next section.

==Shifted log-logistic distribution==

{{redirectstohere|Shifted log-logistic distribution|hide=true}}
{{Probability distribution |
  name       =Shifted log-logistic |
  type       =density|
   pdf_image  =[[Image:Shiftedloglogisticpdf.svg|thumb|275px|<math>\mu=0, \sigma=1,</math> values of <math>\xi</math> as shown in legend]]|
   cdf_image  =[[Image:Shiftedloglogisticcdf.svg|thumb|275px|<math>\mu=0, \sigma=1,</math> values of <math>\xi</math> as shown in legend]]|
  parameters =
<math>\mu \in (-\infty,+\infty) \,</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>\sigma \in (0,+\infty)    \,</math> [[scale parameter|scale]] (real)<br />
<math>\xi\in (-\infty,+\infty)  \,</math> [[shape parameter|shape]] (real)|
  support    =
<math>x \geqslant \mu -\sigma/\xi\,\;(\xi > 0)</math><br />
<math>x \leqslant \mu -\sigma/\xi\,\;(\xi < 0)</math><br/>
<math>x \in (-\infty, +\infty)   \,\;(\xi = 0)</math>|
  pdf        =<math>\frac{(1+\xi z)^{-(1/\xi +1)}}{\sigma\left(1 + (1+\xi z)^{-1/\xi}\right)^2} </math><br />
where <math>z=(x-\mu)/\sigma\,</math>|
  cdf        =<math>\left(1+(1 + \xi z)^{-1/\xi}\right)^{-1} \,</math><br />
where <math>z=(x-\mu)/\sigma\,</math>|
  mean       =<math>\mu + \frac{\sigma}{\xi}(\alpha \csc(\alpha)-1)</math><br />
where <math>\alpha= \pi \xi\, </math>|
  median     =<math>\mu \,</math>|
  mode       =<math>\mu + \frac{\sigma}{\xi}\left[\left(\frac{1-\xi}{1+\xi}\right)^\xi - 1 \right] </math>|
  variance   =<math> \frac{\sigma^2}{\xi^2}[2\alpha \csc(2 \alpha) - (\alpha \csc(\alpha))^2]  </math><br />
where <math>\alpha= \pi \xi\, </math>|
  skewness   =|
  kurtosis   =|
  entropy    =|
  mgf        =|
  char       =|
}}

The '''shifted log-logistic distribution''' is also known as the '''generalized log-logistic''', the '''generalized logistic''',
or the '''three-parameter log-logistic''' distribution.<ref name=Hosking97>{{Citation|title=Regional Frequency Analysis: An Approach Based on L-Moments| first=Jonathan R. M. | last=Hosking| first2= James R| last2=Wallis| year=1997| publisher=Cambridge University Press| isbn=0521430453}}</ref>
<ref>
{{Citation|first=Gary G.|last=Venter| pages=91-101| title=Introduction to selected papers from the variability in reserves prize program | journal=Casualty Actuarial Society Forum |  date=Spring 1994| volume=1| url=http://www.casact.org/pubs/forum/94spforum/94spf091.pdf}}</ref><ref>
{{Citation| first=Ronald B. |last=Geskus |title=Methods for estimating the AIDS incubation time distribution when date of seroconversion is censored| journal=Statistics in Medicine| volume=20| number=5| pages=795-812| year=2001| doi=10.1002/sim.700}}</ref>
It can be obtained from the log-logistic distribution by addition of a shift parameter <math>\delta</math>: if <math>X</math> has a log-logistic distribution then <math>X+\delta</math> has a shifted log-logistic distribution. So  <math>Y</math> has a shifted log-logistic distribution if <math>\log(Y-\delta)</math> 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 [[Pareto distribution#Generalized Pareto distribution|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]] of the shifted log-logistic distribution is
: <math>F(x; \mu,\sigma,\xi) = \frac{1}{ 1 + \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}}</math>
for <math> 1 + \xi(x-\mu)/\sigma \geqslant 0</math>, where <math>\mu\in\mathbb R</math> is the location parameter, <math>\sigma>0\,</math> the scale parameter and <math>\xi\in\mathbb R</math> the shape parameter.  Note that some references use <math> \kappa =  - \xi\,\!</math> to parameterise the shape.<ref name=Hosking97/><ref name=FEH/>

The [[probability density function]] is 
:<math> 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} . </math><br />
again, for <math> 1 + \xi(x-\mu)/\sigma \geqslant 0. </math>

The shape parameter <math>\xi</math> is often restricted to lie in [-1,1], when the probability density function is bounded. When <math>|\xi|>1</math>, it has an [[asymptote]] at <math>x = \mu - \sigma/\xi</math>. Reversing the sign of <math>\xi</math> reflects the pdf and the cdf about <math>x=0.</math>.

===Related distributions===

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

===Applications===

The three-parameter log-logistic distribution is used in [[hydrology]] for modelling  flood frequency.<ref name=Hosking97/><ref name=FEH>
{{Citation|title=Flood Estimation Handbook | volume=3: "Statistical Procedures for Flood Frequency Estimation"| last=Robson| first=A. |last2=Reed| first2=D.| year=1999 |place=Wallingford, UK| publisher= Institute of Hydrology| isbn=0948540893}}</ref><ref>
{{Citation| journal=Journal of Hydrology |volume=98| year=1988 |pages=205-224| title=Log-logistic flood frequency analysis |first=M. I. |last=Ahmad| first2=C. D. |last2=Sinclair |first3=A. |last3=Werritty| doi=10.1016/0022-1694(88)90015-7}}</ref>

==See also==

*[[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|2}}

{{ProbDistributions|continuous-semi-infinite}}

[[Category:Continuous distributions]]
[[Category:Survival analysis]]