WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

LogGammaDistribution
✖
LogGammaDistribution

✖

represents a log-gamma distribution with shape parameters α and β and location parameter μ.

Details

Background & Context

  • LogGammaDistribution[α,β,μ] represents a continuous statistical distribution supported over the interval and parametrized by a non-negative real number μ (called a "location parameter") and by positive real numbers α and β (called "shape parameters") that together determine the overall behavior of its probability density function (PDF). Depending on the values of α and β, the PDF of a log-gamma 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-gamma 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.) Distributions whose densities are proportional to the PDF of ExpGammaDistribution are sometimes mistakenly referred to as LogGammaDistribution, though such distributions can be distinguished from the actual log-gamma distribution by the doubly-exponential behavior of their PDFs.
  • The log-gamma distribution (with zero location parameter) is mathematically defined to be the distribution that models whenever XGammaDistribution. It was shown in a 1971 paper by Consul and Jain that the log-gamma distribution can be used as an approximation tool both to determine the independence of two sets of normally-distributed random variables as well as to test linear hypotheses regarding matrix regression coefficients. The log-gamma distribution can also model a variety of phenomena including income distribution and arrival and departure times in queueing theory, and generalizations thereof have been used as prior distributions in Bayesian analysis to allow for the inclusion of prior knowledge regarding correlations between parameters when likelihood is non-normally distributed.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a log-gamma distribution. Distributed[x,LogGammaDistribution[α,β,μ]], written more concisely as xLogGammaDistribution[α,β,μ], can be used to assert that a random variable x is distributed according to a log-gamma 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-gamma distributions may be given using PDF[LogGammaDistribution[α,β,μ],x] and CDF[LogGammaDistribution[α,β,μ],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-gamma distribution, EstimatedDistribution to estimate a log-gamma parametric distribution from given data, and FindDistributionParameters to fit data to a log-gamma distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic log-gamma distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic log-gamma distribution.
  • TransformedDistribution can be used to represent a transformed log-gamma 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-gamma distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving log-gamma distributions.
  • LogGammaDistribution is related to a number of other distributions. LogGammaDistribution can be realized as a transformation (TransformedDistribution) of GammaDistribution in the sense that the PDF of TransformedDistribution[Log[u+1],uLogGammaDistribution[α,β,0]] is precisely the same as that of PDF[GammaDistribution[α,β],x]. Its logarithmic behavior is also qualitatively similar to that of LogLogisticDistribution, LogMultinormalDistribution, and LogNormalDistribution.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Probability density function:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-7els8w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-rb3efu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-roh2dh
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-g9bdsu
Out[4]=4

Cumulative distribution function:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-rcil56
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-g7daaj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-pos9xx
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-yeqj9a
Out[4]=4

Mean and variance:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-xxpa5e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-inx1yd
Out[2]=2

Median:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-iret27
Out[1]=1

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

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

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-qhtk5j

Compare its histogram to the PDF:

In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-03mwaz
Out[2]=2

Distribution parameters estimation:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-epi747
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-f8ui5o
Out[3]=3

Skewness depends on the shape parameters:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-3pittv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-d00cbn
Out[2]=2

Kurtosis depends on the shape parameters:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-qmp17a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-kq15v
Out[2]=2

Different moments with closed forms as functions of parameters:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-js043h

Moment:

In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-rx074o
Out[2]=2

CentralMoment:

In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-pknsqa
Out[3]=3

FactorialMoment:

In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-zg9ct4
Out[4]=4

Cumulant:

In[5]:=5
✖
https://wolfram.com/xid/0cptf8g3cximqi-9gzmth
Out[5]=5

Hazard function:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-77bxoo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-woi9p6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-cgopjg
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-1ntnhq
Out[4]=4

Quantile function:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-hfoil7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-la105f
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-wkixvv
Out[3]=3

Use dimensionless Quantity to define LogGammaDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-d70hxe
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

Use LogGammaDistribution to model incomes at a large state university:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-wezkil
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-y1qxr5

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

In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-mdtaze

Fit log-gamma distribution to the data:

In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-67cwjj
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0cptf8g3cximqi-jhx4vm
Out[5]=5

Find the average income at the large state university:

In[6]:=6
✖
https://wolfram.com/xid/0cptf8g3cximqi-mq8j3y
Out[6]=6

Find the probability that a salary is at most $25,000:

In[7]:=7
✖
https://wolfram.com/xid/0cptf8g3cximqi-sjjly3
Out[7]=7

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

In[8]:=8
✖
https://wolfram.com/xid/0cptf8g3cximqi-wq7893
Out[8]=8

Find the median salary:

In[9]:=9
✖
https://wolfram.com/xid/0cptf8g3cximqi-ur8ugz
Out[9]=9

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

In[10]:=10
✖
https://wolfram.com/xid/0cptf8g3cximqi-si8x51
Out[10]=10

Properties & Relations  (3)Properties of the function, and connections to other functions

Log-gamma distribution is closed under translating by a positive factor:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-lixbt
Out[1]=1

Relationships to other distributions:

Log-gamma distribution is related to GammaDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-hsuq8g
In[2]:=2
✖
https://wolfram.com/xid/0cptf8g3cximqi-1et4jg
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cptf8g3cximqi-lgm52b
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-u24x8k
Out[4]=4

Transformation from gamma distribution:

In[5]:=5
✖
https://wolfram.com/xid/0cptf8g3cximqi-f0jz74
In[6]:=6
✖
https://wolfram.com/xid/0cptf8g3cximqi-nnngh4
Out[6]=6

Neat Examples  (1)Surprising or curious use cases

PDFs for different β values with CDF contours:

In[1]:=1
✖
https://wolfram.com/xid/0cptf8g3cximqi-io0104
In[4]:=4
✖
https://wolfram.com/xid/0cptf8g3cximqi-ceikus
Out[4]=4
Wolfram Research (2010), LogGammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogGammaDistribution.html (updated 2016).
✖
Wolfram Research (2010), LogGammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogGammaDistribution.html (updated 2016).

Text

Wolfram Research (2010), LogGammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogGammaDistribution.html (updated 2016).

✖
Wolfram Research (2010), LogGammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogGammaDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "LogGammaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/LogGammaDistribution.html.

✖
Wolfram Language. 2010. "LogGammaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/LogGammaDistribution.html.

APA

Wolfram Language. (2010). LogGammaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LogGammaDistribution.html

✖
Wolfram Language. (2010). LogGammaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LogGammaDistribution.html

BibTeX

@misc{reference.wolfram_2025_loggammadistribution, author="Wolfram Research", title="{LogGammaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/LogGammaDistribution.html}", note=[Accessed: 06-June-2025 ]}

✖
@misc{reference.wolfram_2025_loggammadistribution, author="Wolfram Research", title="{LogGammaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/LogGammaDistribution.html}", note=[Accessed: 06-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_loggammadistribution, organization={Wolfram Research}, title={LogGammaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/LogGammaDistribution.html}, note=[Accessed: 06-June-2025 ]}

✖
@online{reference.wolfram_2025_loggammadistribution, organization={Wolfram Research}, title={LogGammaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/LogGammaDistribution.html}, note=[Accessed: 06-June-2025 ]}