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In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a
continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
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 Mortality rate, or death rate, is a measure of the number of deaths (in general, or due to a specific cause) in a particular population, scaled to the size of that population, per unit of time. Mortality rate is typically expressed in units of de ...
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 The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or heterogeneity in economics, economic heterogeneity. The distribution of wealth differs from the i ...
or income, and in 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 may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...
. It is similar in shape to the
log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
but has heavier tails. Unlike the log-normal, its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
can be written in 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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
. The parameter \alpha>0 is a scale parameter and is also the median of the distribution. The parameter \beta>0 is a shape parameter. The distribution is
unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal ...
when \beta>1 and its
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
decreases as \beta increases. The
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
is :\begin F(x; \alpha, \beta) & = \\ pt & = \\ pt & = \end where x>0, \alpha>0, \beta>0. The probability density function is :f(x; \alpha, \beta) = \frac


Alternative parameterization

An alternative parametrization is given by the pair \mu, s in analogy with the logistic distribution: : \mu = \ln (\alpha) : s = 1 / \beta


Properties


Moments

The kth raw
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
exists only when k<\beta, when it is given by :\begin \operatorname(X^k) & = \alpha^k\operatorname(1-k/\beta, 1+k/\beta) \\ pt& = \alpha^k\, \end where B is the
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^( ...
. Expressions for the mean, variance,
skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...
and
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtos ...
can be derived from this. Writing b=\pi/\beta for convenience, the mean is : \operatorname(X) = \alpha b / \sin b , \quad \beta>1, and the variance is : \operatorname(X) = \alpha^2 \left( 2b / \sin 2b -b^2 / \sin^2 b \right), \quad \beta>2. Explicit expressions for the skewness and kurtosis are lengthy. As \beta 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^(p;\alpha, \beta) = \alpha\left( \frac \right)^. It follows that the median is \alpha, the lower quartile is 3^ \alpha and the upper quartile is 3^ \alpha.


Applications


Survival analysis

The log-logistic distribution provides one parametric model for survival analysis. Unlike the more commonly used
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...
, it can have a non-
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
hazard function Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering. The failure rate of a ...
: when \beta>1, the hazard function is
unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal ...
(when \beta ≤ 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 In the statistical area of survival analysis, an accelerated failure time model (AFT model) is a parametric model that provides an alternative to the commonly used proportional hazards models. Whereas a proportional hazards model assumes that t ...
by allowing \alpha to differ between groups, or more generally by introducing covariates that affect \alpha but not \beta by modelling \log(\alpha) as a linear function of the covariates. The survival function is :S(t) = 1 - F(t) = +(t/\alpha)^,\, and so the
hazard function Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering. The failure rate of a ...
is : h(t) = \frac = \frac . The log-logistic distribution with shape parameter \beta = 1 is the marginal distribution of the inter-times in a geometric-distributed
counting process A counting process is a stochastic process with values that are non-negative, integer, and non-decreasing: # ''N''(''t'') ≥ 0. # ''N''(''t'') is an integer. # If ''s'' ≤ ''t'' then ''N''(''s'') ≤ ''N''(''t''). If ''s'' < ''t'', then ''N''(' ...
.


Hydrology

The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation. Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a
log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
. 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 In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
based on the
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no questi ...
. The rainfall data are represented by the
plotting position Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot'' ...
''r''/(''n''+1) as part of the
cumulative frequency analysis Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called ''frequency of non-exceedance ...
.


Economics

The log-logistic has been used as a simple model of the
distribution of wealth The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or heterogeneity in economics, economic heterogeneity. The distribution of wealth differs from the i ...
or income in economics, where it is known as the Fisk distribution. Its Gini coefficient is 1/\beta. The Gini coefficient for a continuous probability distribution takes the form: :G = \int_^F(1-F)dx where F is the CDF of the distribution and \mu is the expected value. For the log-logistic distribution, the formula for the Gini coefficient becomes: :G = \int_^ Defining the substitution z = x/\alpha leads to the simpler equation: :G = \int_^ And making the substitution u = 1/(1 + z^) further simplifies the Gini coefficient formula to: :G = \int_^u^(1-u)^du The integral component is equivalent to the standard
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^( ...
\text(1-1/\beta,1+1/\beta). The beta function may also be written as: :\text(x,y) = where \Gamma(\cdot) is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...
. Using the properties of the gamma function, it can be shown that: :\text(1-1/\beta,1+1/\beta) = \Gamma(1-1/\beta)\Gamma(1/\beta) From Euler's reflection formula, the expression can be simplified further: :\text(1-1/\beta,1+1/\beta) = Finally, we may conclude that the Gini coefficient for the log-logistic distribution G = 1/\beta.


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 travelling through and being processed by other computers, applications, and network segments, most or all of them without hard
real-time Real-time or real time describes various operations in computing or other processes that must guarantee response times within a specified time (deadline), usually a relatively short time. A real-time process is generally one that happens in defined ...
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 In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
or others, as long as abrupt changes of regime in the sequences of those times are properly detected.


Related distributions

* If X \sim LL(\alpha,\beta) then kX \sim LL(k \alpha, \beta). * If X \sim LL(\alpha, \beta) then X^k \sim LL(\alpha^k, \beta/, k, ). * LL(\alpha,\beta) \sim \textrm(1,\alpha,\beta) ( Dagum distribution). * LL(\alpha,\beta) \sim \textrm(1,\alpha,\beta) ( Singh–Maddala distribution). * \textrm(\gamma,\sigma) \sim \beta'(1,1,\gamma,\sigma) (
Beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions ...
). *If ''X'' has a log-logistic distribution with scale parameter \alpha and shape parameter \beta then ''Y'' = log(''X'') has a
logistic distribution Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...
with location parameter \log(\alpha) and scale parameter 1/\beta. *As the shape parameter \beta of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow)
logistic distribution Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...
. Informally: ::LL(\alpha, \beta) \to L(\alpha,\alpha/\beta) \quad \text \quad \beta \to \infty. *The log-logistic distribution with shape parameter \beta=1 and scale parameter \alpha is the same as the
generalized Pareto distribution In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location \mu, scale \sigma, and shap ...
with location parameter \mu=0, shape parameter \xi=1 and scale parameter \alpha: ::LL(\alpha,1) = GPD(1,\alpha,1). *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. These include the
Burr Type XII distribution In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala 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 The metalog distribution is a flexible continuous probability distribution designed for ease of use in practice. Together with its transforms, the metalog family of continuous distributions is unique because it embodies ''all'' of following proper ...
, in which power series expansions in terms of p are substituted for
logistic distribution Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...
parameters \mu and \sigma. The resulting log-metalog distribution is highly shape flexible, has simple closed form
PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems ...
and quantile function, can be fit to data with linear least squares, and subsumes the log-logistic distribution is special case.


See also

* Probability distributions: List of important distributions supported on semi-infinite intervals


References

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