LogLogisticDistribution
represents a loglogistic distribution with shape parameter γ and scale parameter σ.
Details
 LogLogisticDistribution is also known as Fisk distribution.
 The probability density for value in a loglogistic distribution is proportional to for .
 LogLogisticDistribution allows γ and σ to be any positive real numbers.
 LogLogisticDistribution allows σ to be a quantity of any unit dimension, and γ to be a dimensionless quantity. »
 LogLogisticDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
 LogLogisticDistribution[γ,σ] represents a continuous statistical distribution supported over the interval and parametrized by positive real numbers γ (called a "shape parameter") and σ (called a "scale parameter"), which together determine the overall behavior of its probability density function (PDF). Depending on the values of γ and σ, the PDF of a loglogistic distribution may be either unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with a potential singularity approaching the lower boundary of its domain. In addition, the PDF of the loglogistic distribution has tails that are "fat" in the sense that its PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) LogLogisticDistribution is sometimes referred to as the Fisk distribution, particularly in applications to economics.
 LogLogisticDistribution is the distribution followed by the logarithm of a logisticdistributed random variable. In other words, if is a random variable and (where denotes "is distributed as"), then . Qualitatively, the loglogistic distribution is similar to the lognormal distribution (LogNormalDistribution) and as such, both are commonly utilized tools for approximating lifetime data across various disciplines. In addition, the loglogistic distribution has been employed to model numerous phenomena including precipitation, wealth and income distribution, and data transmission and processing times.
 RandomVariate can be used to give one or more machine or arbitraryprecision (the latter via the WorkingPrecision option) pseudorandom variates from a loglogistic distribution. Distributed[x,LogLogisticDistribution[γ,σ]], written more concisely as xLogLogisticDistribution[γ,σ], can be used to assert that a random variable x is distributed according to a loglogistic distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
 The probability density and cumulative distribution functions for loglogistic distributions may be given using PDF[LogLogisticDistribution[γ,σ],x] and CDF[LogLogisticDistribution[γ,σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
 DistributionFitTest can be used to test if a given dataset is consistent with a loglogistic distribution, EstimatedDistribution to estimate a loglogistic parametric distribution from given data, and FindDistributionParameters to fit data to a loglogistic distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic loglogistic distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic loglogistic distribution.
 TransformedDistribution can be used to represent a transformed loglogistic distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higherdimensional distributions that contain a loglogistic distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving loglogistic distributions.
 LogLogisticDistribution is related to a number of other distributions. LogLogisticDistribution is connected to LogisticDistribution and is qualitatively similar to LogNormalDistribution as discussed above. LogLogisticDistribution is a special case of several distributions including DagumDistribution, SinghMaddalaDistribution, and BetaPrimeDistribution in the sense that the PDF of LogLogisticDistribution[γ,σ] is precisely the same as that of DagumDistribution[1,γ,σ], SinghMaddalaDistribution[1,γ,σ], and BetaPrimeDistribution[1,1,γ,σ]. Its logarithmic behavior is qualitatively similar to that of LogGammaDistribution, LogMultinormalDistribution, and LogNormalDistribution. LogLogisticDistribution is also related to DavisDistribution, NormalDistribution, ExponentialDistribution, WeibullDistribution, GompertzMakehamDistribution, and GammaDistribution.
Examples
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Scope (8)
Applications (2)
Properties & Relations (6)
Neat Examples (1)
Introduced in 2010
Updated in 2016
(8.0)

(10.4)