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SinghMaddalaDistribution
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SinghMaddalaDistribution

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represents the Singh–Maddala distribution with shape parameters q and a and scale parameter b.

Details

Background & Context

  • SinghMaddalaDistribution[q,a,b] represents a continuous statistical distribution supported on the interval and parametrized by positive real numbers q, a, and b (two "shape parameters" and a "scale parameter", respectively) that together determine the overall behavior of its probability density function (PDF). Depending on the values of q, a, and b, the PDF of a Singh–Maddala distribution may have any of a number of shapes including unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with potential singularities approaching the lower boundary of its domain. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Singh–Maddala distribution is also sometimes referred to as the Burr XII distribution or as the Burr distribution and is one of a number of distributions referred to as generalized log-logistic distributions (not to be confused with LogLogisticDistribution).
  • The Singh–Maddala distribution was first discovered in the early 1940s by I. W. Burr before being rediscovered in the 1970s by S. K. Singh and G. S. Maddala as an alternative to the gamma (GammaDistribution) and log-normal distributions (LogNormalDistribution) in modeling income distribution. Since then, the Singh–Maddala distribution has been used ubiquitously throughout economics and econometrics to model various financial phenomena and has also been used as a tool in areas such as actuarial science, Monte Carlo theory, publishing, and sociology.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Singh–Maddala distribution. Distributed[x,SinghMaddalaDistribution[q,a,b]], written more concisely as xSinghMaddalaDistribution[q,a,b], can be used to assert that a random variable x is distributed according to a Singh–Maddala 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 Singh–Maddala distributions may be given using PDF[SinghMaddalaDistribution[q,a,b],x] and CDF[SinghMaddalaDistribution[q,a,b],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 Singh–Maddala distribution, EstimatedDistribution to estimate a Singh–Maddala parametric distribution from given data, and FindDistributionParameters to fit data to a Singh–Maddala distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Singh–Maddala distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Singh–Maddala distribution.
  • TransformedDistribution can be used to represent a transformed Singh–Maddala 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 Singh–Maddala distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Singh–Maddala distributions.
  • SinghMaddalaDistribution is related to a number of other distributions. SinghMaddalaDistribution generalizes LogLogisticDistribution (the PDF of SinghMaddalaDistribution[1,γ,σ] is precisely that of LogLogisticDistribution[γ,σ]), is generalized by BetaPrimeDistribution (the PDF of SinghMaddalaDistribution[q,a,b] is exactly that of BetaPrimeDistribution[1,q,a,b]), and is a transformation (TransformedDistribution) of DagumDistribution. SinghMaddalaDistribution is also closely related to BetaDistribution, ParetoDistribution, PearsonDistribution, GammaDistribution, WeibullDistribution, 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/0bcmxv4bupxw1ziuhx7p2-dp560l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-0mhod
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-okbt41
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-p8y9kn
Out[4]=4

Cumulative distribution function:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-v8nqua
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-0ssfqe
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ug225g
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-c137pl
Out[4]=4

Mean and variance may not be defined for all parameter values:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-wa7dmd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ymdvcr
Out[2]=2

Median:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-jii59n
Out[1]=1

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

Generate a sample of pseudorandom numbers from a Singh–Maddala distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-qhtk5j

Compare its histogram to the PDF:

In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-03mwaz
Out[2]=2

Distribution parameters estimation:

In[1]:=1
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-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/0bcmxv4bupxw1ziuhx7p2-f8ui5o
Out[3]=3

Skewness varies with the shape parameters and and is defined when :

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-bjfi3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-uf1tuq
Out[2]=2

Kurtosis is defined when :

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-035hnx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-na13lv
Out[2]=2

Different moments with closed forms as functions of parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-js043h

Moment:

In[2]:=2
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-rx074o
Out[2]=2

Closed form for symbolic order:

In[3]:=3
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-688274
Out[3]=3

CentralMoment:

In[4]:=4
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-pknsqa
Out[4]=4

FactorialMoment:

In[5]:=5
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-zg9ct4
Out[5]=5

Cumulant:

In[6]:=6
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-9gzmth
Out[6]=6

Hazard function:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-l0tz6m
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-pbd2t1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-1cfb8x
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-kqzsl3
Out[4]=4

Quantile function:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-28o0no
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-4jhunk
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ggehhe
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ryatvn
Out[4]=4

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-bc5g3d
Out[1]=1

Find the quartiles:

In[2]:=2
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https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ep3kcu
Out[2]=2

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

The number of earthquakes per year can be modeled with SinghMaddalaDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-2nwq5w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ehjcp4
Out[2]=2

Fit the distribution to the data:

In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-bk6atw
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-p3pxb2
Out[4]=4

Find the probability of at least 60 earthquakes in the US in a year:

In[5]:=5
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-zysa5u
Out[5]=5

Find the mean amount of earthquakes in the US in a year:

In[6]:=6
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-qks7dt
Out[6]=6

Simulate the number of earthquakes per year for the next 30 years:

In[7]:=7
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-zjxofb
Out[7]=7

Properties & Relations  (8)Properties of the function, and connections to other functions

Singh–Maddala distribution is closed under scaling by a positive factor:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-cvj157
Out[1]=1

The family of SinghMaddalaDistribution is closed under a minimum:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-xm1acx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-pz5gta
Out[2]=2

The hazard function is unimodal for , and decreasing for :

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-nojjji
Out[1]=1

The parameter q is a scale factor for the hazard function:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-iqxm2m
Out[1]=1

Relations to other distributions:

SinghMaddalaDistribution is a special case of BetaPrimeDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-dosmau
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-6xu54b
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-8qecm
Out[3]=3

If has a SinghMaddalaDistribution, then has a DagumDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ztevh0
Out[1]=1

LogLogisticDistribution is a special case of SinghMaddalaDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-c1jcmn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-d26l8s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-38spiv
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

PDFs for different q values with CDF contours:

In[1]:=1
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-io0104
In[4]:=4
✖
https://wolfram.com/xid/0bcmxv4bupxw1ziuhx7p2-ceikus
Out[4]=4
Wolfram Research (2010), SinghMaddalaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SinghMaddalaDistribution.html (updated 2016).
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Wolfram Research (2010), SinghMaddalaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SinghMaddalaDistribution.html (updated 2016).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_singhmaddaladistribution, author="Wolfram Research", title="{SinghMaddalaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/SinghMaddalaDistribution.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_singhmaddaladistribution, author="Wolfram Research", title="{SinghMaddalaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/SinghMaddalaDistribution.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_singhmaddaladistribution, organization={Wolfram Research}, title={SinghMaddalaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/SinghMaddalaDistribution.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_singhmaddaladistribution, organization={Wolfram Research}, title={SinghMaddalaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/SinghMaddalaDistribution.html}, note=[Accessed: 29-March-2025 ]}