LogLogisticDistribution

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`LogLogisticDistribution`

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LogLogisticDistribution[γ,σ]

represents a log-logistic distribution with shape parameter γ and scale parameter σ.

Details

LogLogisticDistribution is also known as Fisk distribution.
The probability density for value in a log-logistic 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 log-logistic 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 log-logistic 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 logistic-distributed random variable. In other words, if is a random variable and (where denotes "is distributed as"), then . Qualitatively, the log-logistic distribution is similar to the log-normal distribution (LogNormalDistribution) and as such, both are commonly utilized tools for approximating lifetime data across various disciplines. In addition, the log-logistic 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 arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a log-logistic distribution. Distributed[x,LogLogisticDistribution[γ,σ]], written more concisely as xLogLogisticDistribution[γ,σ], can be used to assert that a random variable x is distributed according to a log-logistic 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 log-logistic 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 log-logistic distribution, EstimatedDistribution to estimate a log-logistic parametric distribution from given data, and FindDistributionParameters to fit data to a log-logistic distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic log-logistic distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic log-logistic distribution.
TransformedDistribution can be used to represent a transformed log-logistic 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 higher-dimensional distributions that contain a log-logistic distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving log-logistic 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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Basic Examples (4)Summary of the most common use cases

Probability density function:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-mm0u45

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-lzvs1w

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ud24zr

Out[3]=3

Cumulative distribution function:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-k8opug

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-sl6mbg

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-t1dyqi

Out[3]=3

Mean and variance:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-pkjpxb

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-4b9ibt

Out[2]=2

Median:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-6kedsx

Out[1]=1

Scope (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a log-logistic distribution:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-qhtk5j

Compare its histogram to the PDF:

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-03mwaz

Out[2]=2

Distribution parameters estimation:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-epi747

Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-f8ui5o

Out[3]=3

Skewness depends only on the shape parameter γ:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-s2le6s

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-h5v6ql

Out[2]=2

For large values of γ, log-logistic distribution becomes symmetric:

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-8w817q

Out[3]=3

Kurtosis depends only on the shape parameter γ:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-o0xwfi

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-0vcsns

Out[2]=2

Kurtosis has horizontal asymptote as γ gets large:

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-tcrg9l

Out[3]=3

Different moments with closed forms as functions of parameters:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-js043h

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-rx074o

Out[2]=2

Closed form for symbolic order:

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-vdnhk1

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-pknsqa

Out[4]=4

FactorialMoment:

In[5]:=5

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-zg9ct4

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-9gzmth

Out[6]=6

Hazard function:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-2iq3ge

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-5hz33a

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ilrg2w

Out[3]=3

Quantile function:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-x3ve33

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-iml77m

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-sdhku4

Out[3]=3

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-bc5g3d

Out[1]=1

Find the interquartile range:

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ep3kcu

Out[2]=2

Applications (2)Sample problems that can be solved with this function

LogLogisticDistribution can be used to model incomes:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-5mvzlk

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-87moqy

Adjust part-time to full-time and select nonzero values, attach currency unit:

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-w8fgub

Fit log-logistic distribution to the data:

In[4]:=4

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-83xl39

Out[4]=4

Compare the histogram of the data to the PDF of the estimated distribution:

In[5]:=5

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-wrn7k9

Out[5]=5

Find the average income at a large state university:

In[6]:=6

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-for81j

Out[6]=6

Find the probability that a salary is at most $15,000:

In[7]:=7

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-fl05eu

Out[7]=7

Find the probability that a salary is at least $150,000:

In[8]:=8

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-mzibvb

Out[8]=8

Find the median salary:

In[9]:=9

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-7hcsn8

Out[9]=9

Simulate the incomes for 100 randomly selected employees of such a university:

In[10]:=10

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-18pcw5

Out[10]=10

BetaPrimeDistribution can be used to model state per capita incomes:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-bdhhru

Out[1]=1

Attach currency units:

In[2]:=2

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-zrq5bc

Fit log-logistic distribution to the data:

In[3]:=3

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-24wzdp

Out[3]=3

Compare the histogram of the data to the PDF of the estimated distribution:

In[4]:=4

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-4l4py7

Out[4]=4

Find the average income per capita:

In[5]:=5

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-belxf6

Out[5]=5

Find states with income close to the average:

In[6]:=6

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-uen50d

Out[6]=6

Find the median income per capita:

In[7]:=7

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-wbveuy

Out[7]=7

Find states with income close to the median:

In[8]:=8

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-0mngys

Out[8]=8

Find log-likelihood value:

In[9]:=9

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-w0ag8d

Out[9]=9

Properties & Relations (6)Properties of the function, and connections to other functions

Log-logistic distribution is closed under scaling by a positive factor:

In[1]:=1

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-u1rz2n

Out[1]=1

Relationships to other distributions:

LogLogisticDistribution is a special case of DagumDistribution:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-s5fo3i

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-gzciyh

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-bmd664

Out[3]=3

LogLogisticDistribution is a special case of SinghMaddalaDistribution:

In[1]:=1

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https://wolfram.com/xid/0nx3ljk9ysmq9gk86-c1jcmn

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-d26l8s

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-38spiv

Out[3]=3

LogLogisticDistribution is a special case of BetaPrimeDistribution:

In[1]:=1

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-h0l32b

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-n117d9

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-clrien

Out[3]=3

LogLogisticDistribution is related to LogisticDistribution:

In[1]:=1

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-6cu6y0

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-1p3yh7

Out[2]=2

Neat Examples (1)Surprising or curious use cases

PDFs for different γ values with CDF contours:

In[1]:=1

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-io0104

In[4]:=4

✖

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ceikus

Out[4]=4

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