Log-logistic distribution


Log-logistic distribution

Probability distribution
name =Log-logistic
type =density
pdf_

cdf_

parameters =alpha>0 scale eta> 0 shape
support =xin [0,infty)
pdf = frac{ (eta/alpha)(x/alpha)^{eta-1} } { left [ 1+(x/alpha)^{eta} ight] ^2 }
cdf ={ 1 over 1+(x/alpha)^{-eta} }
mean ={alpha,pi/eta over sin(pi/eta)} if eta>1, else undefined
median =alpha,
mode =alphaleft(frac{eta-1}{eta+1} ight)^{1/eta} if eta> 1, 0 otherwise
variance =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.

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, 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.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] 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] The parameter alpha>0 is a scale parameter and is also the median of the distribution. The parameter eta>0 is a shape parameter. The distribution is unimodal when eta>1 and its dispersion decreases as eta increases.

The cumulative distribution function is:egin{align} F(x; alpha, eta) & = { 1 over 1+(x/alpha)^{-eta} } \ & = {(x/alpha)^eta over 1+(x/alpha)^ eta } \ & = {x^eta over alpha^eta+x^eta}end{align}where x>0, alpha>0, eta>0.

The probability density function is:f(x; alpha, eta) = frac{ (eta/alpha)(x/alpha)^{eta-1} } { left [ 1+(x/alpha)^{eta} ight] ^2 }.

Properties

Moments

The kth raw moment exists only when k<eta, when it is given byCitation| 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] 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] :egin{align}operatorname{E}(X^k) & = alpha^k,operatorname{B}(1-k/eta,, 1+k/eta) \& = alpha^k, {k,pi/eta over sin(k,pi/eta)}end{align} where B() is the beta function.Expressions for the mean, variance, skewness and kurtosis can be derived from this. Writing b=pi/eta for convenience, the mean is: operatorname{E}(X) = alpha b / sin b , quad eta>1,and the variance is: operatorname{Var}(X) = alpha^2 left( 2b / sin 2b -b^2 / sin^2 b ight), quad eta>2.Explicit expressions for the kurtosis and variance are lengthy. [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] As eta tends to infinity the mean tends to alpha, the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below).

Quantiles

The quantile function (inverse cumulative distribution function) is ::F^{-1}(p;alpha, eta) = alphaleft( frac{p}{1-p} ight)^{1/eta}.It follows that the median is alpha, the lower quartile is 3^{1/eta} alpha and the upper quartile is 3^{-1/eta} alpha.

Applications

urvival 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 eta>1, the hazard function is unimodal (when eta ≤ 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. [] The log-logistic distribution can be used as the basis of an accelerated failure time model by allowing eta to differ between groups, or more generally by introducing covariates that affect eta but not alpha by modelling log(eta) as a linear function of the covariates. [Citation | title =Modelling Survival Data in Medical Research|first=Dave |last=Collett | year=2003 | edition=2nd | publisher=CRC press| isbn=1584883251]

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

Hydrology

The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation.

Economics

The log-logistic has been used as a simple model of the distribution of wealth or income in economics, where it is known as the Fisk distribution. [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] Its Gini coefficient is 1/eta.Citation|last1=Kleiber |first1=C. |last2=Kotz| first2= S| year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences | publisher=Wiley | isbn=0471150649]

Related distributions

*If "X" has a log-logistic distribution with scale parameter alpha and shape parameter eta then "Y" = log("X") has a logistic distribution with location parameter log(alpha) and scale parameter eta.

*As the shape parameter eta of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow) logistic distribution. Informally, as eta→∞, :LL(alpha, eta) o L(alpha,alpha/eta).

*The log-logistic distribution with shape parameter eta=1 and scale parameter alpha is the same as the generalized Pareto distribution with location parameter mu=0, shape parameter xi=1 and scale parameter alpha::LL(alpha,1) = GPD(1,alpha,1).,

Generalizations

Several different distributions are sometimes referred to as the generalized log-logistic distribution, as they contain the log-logistic as a special case. 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.

hifted log-logistic distribution

Probability distribution
name =Shifted log-logistic
type =density
pdf_

cdf_

parameters =mu in (-infty,+infty) , location (real)
sigma in (0,+infty) , scale (real)
xiin (-infty,+infty) , shape (real)
support =x geqslant mu -sigma/xi,;(xi > 0)
x leqslant mu -sigma/xi,;(xi < 0) x in (-infty, +infty) ,;(xi = 0)
pdf =frac{(1+xi z)^{-(1/xi +1)
{sigmaleft(1 + (1+xi z)^{-1/xi} ight)^2}
where z=(x-mu)/sigma,
cdf =left(1+(1 + xi z)^{-1/xi} ight)^{-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} ight)^xi - 1 ight]
variance = frac{sigma^2}{xi^2} [2alpha csc(2 alpha) - (alpha csc(alpha))^2]
where alpha= pi xi,
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.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] [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] [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] It can be obtained from the log-logistic distribution by addition of a shift parameter delta: 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 of the shifted log-logistic distribution is: F(x; mu,sigma,xi) = frac{1}{ 1 + left(1+ frac{xi(x-mu)}{sigma} ight)^{-1/xifor 1 + xi(x-mu)/sigma geqslant 0, where muinmathbb R is the location parameter, sigma>0, the scale parameter and xiinmathbb R the shape parameter. Note that some references use kappa = - xi,! to parameterise the shape.

The probability density function is : f(x; mu,sigma,xi) = frac{left(1+frac{xi(x-mu)}{sigma} ight)^{-(1/xi +1){sigmaleft [1 + left(1+frac{xi(x-mu)}{sigma} ight)^{-1/xi} ight] ^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=0..

Related distributions

* 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.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] [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]

ee also

*

References


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