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{{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 a [[probability distribution]] also known as the '''generalized log-logistic''' or the '''three-parameter log-logistic''' distribution.<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| pages=795–812| year=2001| doi=10.1002/sim.700| issue=5| pmid=11241577}}</ref> It has also been called the '''generalized 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=0-521-43045-3}}</ref> 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 [[Location parameter|shift parameter]] <math>\delta</math>. Thus 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 distributions|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>
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=\mu.</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=0-948540-89-3}}</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| issue=3–4}}</ref>
 
==References==
{{reflist}}
 
{{ProbDistributions|continuous-variable}}
 
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Latest revision as of 17:56, 18 June 2013

Template:Probability distribution

The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution.[1][2] It has also been called the generalized logistic distribution,[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 δ. Thus if X has a log-logistic distribution then X+δ has a shifted log-logistic distribution. So Y has a shifted log-logistic distribution if log(Yδ) 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 of the shifted log-logistic distribution is

F(x;μ,σ,ξ)=11+(1+ξ(xμ)σ)1/ξ

for 1+ξ(xμ)/σ0, where μ is the location parameter, σ>0 the scale parameter and ξ the shape parameter. Note that some references use κ=ξ to parameterise the shape.[3][4]

The probability density function is

f(x;μ,σ,ξ)=(1+ξ(xμ)σ)(1/ξ+1)σ[1+(1+ξ(xμ)σ)1/ξ]2,

again, for 1+ξ(xμ)/σ0.

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

Applications

The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency.[3][4][5]

References

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