gamma distribution

5/28/2018 Gamma Distribution

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Gamma distribution 1

Gamma distribution

Gamma

Probability density function

Cumulative distribution function

Parameters k> 0 shape

> 0 scale

> 0 shape

> 0 rate

Support

Probability density function (pdf)

Cumulative distribution function (CDF)

Mean

(see digamma function) (see digamma function)

Median No simple closed form No simple closed form

Mode

Variance

(see trigamma function) (see trigamma function)

Skewness

Excess kurtosis
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Entropy

Moment-generating function (mgf)

Characteristic function

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability

distributions. There are three different parametrizations in common use:

1. With a shape parameter k and a scale parameter .

2. With a shape parameter = kand an inverse scale parameter = 1/, called a rate parameter.

3. With a shape parameter k and a mean parameter = k/.

In each of these three forms, both parameters are positive real numbers.

The parameterization with k and appears to be more common in econometrics and certain other applied fields,

where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting

time until death is a random variable that is frequently modeled with a gamma distribution.[1]

The parameterization with and is more common in Bayesian statistics, where the gamma distribution is used as a

conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the of an exponential

distribution or a Poisson distribution or for that matter, the of the gamma distribution itself. (The closely related

inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal

distribution.)

If k is an integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially

distributed random variables, each of which has a mean of (which is equivalent to a rate parameter of 1/).

The gamma distribution is the maximum entropy probability distribution for a random variable X for which E[X] =

k = / is fixed and greater than zero, and E[ln(X)] = (k) + ln() = () ln() is fixed ( is the digamma

function).
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Characterization using shapek and scale

A random variableX that is gamma-distributed with shape k and scale is denoted


Illustration of the gamma PDF for parameter values over k andxwith set to

1, 2, 3, 4, 5 and 6. One can see each layer by itself here [2] as well as by k[3] andx.

[4].

The probability density function using

the shape-scale parametrization is

Here (k) is the gamma function

evaluated at k.

Cumulative distribution

function

The cumulative distribution function is

the regularized gamma function:

where (k, x/) is the lower incomplete

gamma function.

It can also be expressed as follows, if k

is a positive integer (i.e., the

distribution is an Erlang distribution):[5]

Characterization using shape and rate

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter = kand an inverse scale

parameter = 1/, called a rate parameter. A random variable X that is gamma-distributed with shape and rate is

denoted


The corresponding density function in the shape-rate parametrization is

Both parametrizations are common because either can be more convenient depending on the situation.
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Cumulative distribution function

The cumulative distribution function is the regularized gamma function:

where (, x) is the lower incomplete gamma function.

If is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the

following series expansion:

Properties

Skewness

The skewness is equal to , it depends only on the shape parameter (k) and approaches a normal distribution

when k is large (approximately when k > 10).

Median calculation

Unlike the mode and the mean which have readily calculable formulas based on the parameters, the median does not

have an easy closed form equation. The median for this distribution is defined as the value such that

A formula for approximating the median for any gamma distribution, when the mean is known, has been derived

based on the fact that the ratio /( ) is approximately a linear function of k when k 1.[6]

The approximation

formula is

where is the mean.

Summation

IfXihas a Gamma(k

i, ) distribution for i= 1, 2, ...,N (i.e., all distributions have the same scale parameter ), then

provided allXiare independent.

For the cases where theXiare independent but have different scale parameters see Mathai (1982) and Moschopoulos

(1984).

The gamma distribution exhibits infinite divisibility.
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Scaling

If

then for any c> 0,

Hence the use of the term "scale parameter" to describe .

Equivalently, if

then for any c> 0,

Hence the use of the term "inverse scale parameter" to describe .

Exponential family

The gamma distribution is a two-parameter exponential family with natural parameters k1 and 1/ (equivalently,

1 and ), and natural statisticsX and ln(X).

If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential

family.

Logarithmic expectation

One can show that

or equivalently,

where is the digamma function.

This can be derived using the exponential family formula for the moment generating function of the sufficient

statistic, because one of the sufficient statistics of the gamma distribution is ln(x).
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Information entropy

The information entropy is

In the k, parameterization, the information entropy is given by

KullbackLeibler divergence

Illustration of the KullbackLeibler (KL) divergence for two gamma PDFs. Here

= 0+ 1 which are set to 1, 2, 3, 4, 5 and 6. The typical asymmetry for the KL

divergence is clearly visible.

The KullbackLeibler divergence

(KL-divergence), as with the

information entropy and various other

theoretical properties, are more

commonly seen using the ,

parameterization because of their uses

in Bayesian and other theoretical

statistics frameworks.

The KL-divergence of Gamma(p

, p

)

("true" distribution) from Gamma(q,

q) ("approximating" distribution) is

given by[7]

Written using the k, parameterization, the KL-divergence of Gamma(kp

, p

) from Gamma(kq,

q) is given by

Laplace transform

The Laplace transform of the gamma PDF is

Parameter estimation

Maximum likelihood estimation

The likelihood function forNiid observations (x1, ...,x

N) is

from which we calculate the log-likelihood function
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Finding the maximum with respect to by taking the derivative and setting it equal to zero yields the maximum

likelihood estimator of the parameter:

Substituting this into the log-likelihood function gives

Finding the maximum with respect to k by taking the derivative and setting it equal to zero yields

There is no closed-form solution for k. The function is numerically very well behaved, so if a numerical solution is

desired, it can be found using, for example, Newton's method. An initial value of k can be found either using themethod of moments, or using the approximation

If we let

then k is approximately

which is within 1.5% of the correct value.[8]

An explicit form for the Newton-Raphson update of this initial guess

is:[9]

Bayesian minimum mean-squared error

With known k and unknown , the posterior density function for theta (using the standard scale-invariant prior for )

is

Denoting

Integration over can be carried out using a change of variables, revealing that 1/ is gamma-distributed with

parameters =Nk, =y.

The moments can be computed by taking the ratio (m by m = 0)
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which shows that the mean standard deviation estimate of the posterior distribution for theta is

Generating gamma-distributed random variables

Given the scaling property above, it is enough to generate gamma variables with = 1 as we can later convert to any

value of with simple division.

Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of

generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then ln(U) is distributed

Gamma(1, 1) Now, using the "-addition" property of gamma distribution, we expand this result:

where Uk

are all uniformly distributed on (0, 1] and independent. All that is left now is to generate a variable

distributed as Gamma(, 1) for 0 < < 1 and apply the "-addition" property once more. This is the most difficult

part.

Random generation of gamma variates is discussed in detail by Devroye,[10]

noting that none are uniformly fast for

all shape parameters. For small values of the shape parameter, the algorithms are often not valid.[11]

For arbitrary

values of the shape parameter, one can apply the Ahrens and Dieter[12]

modified acceptance-rejection method

Algorithm GD (shape k 1), or transformation method when 0 < k< 1. Also see Cheng and Feast Algorithm GKM

3[13]

or Marsaglia's squeeze method.[14]

The following is a version of the Ahrens-Dieter acceptance-rejection method:

1. Let m be 1.2. Generate V

3m2, V

3m1and V

3mas independent uniformly distributed on (0, 1] variables.

3. If , where , then go to step 4, else go to step 5.

4. Let . Go to step 6.

5. Let .

6. If , then increment m and go to step 2.

7. Assume = m

to be the realization of (, 1).

A summary of this is

where

is the integral part of k,

has been generated using the algorithm above with = {k} (the fractional part of k),

Uk

and Vlare distributed as explained above and are all independent.

While the above approach is technically correct, Devroye notes that it is linear in the value of k and in general is not

a good choice. Instead he recommends using either rejection-based or table-based methods, depending on context.[15]
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Related distributions

Conjugate prior

In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: the Poisson,

exponential, normal (with known mean), Pareto, gamma with known shape , inverse gamma with known shape

parameter, and Gompertz with known scale parameter.

The gamma distribution's conjugate prior is:[16]

whereZ is the normalizing constant, which has no closed-form solution. The posterior distribution can be found by

updating the parameters as follows:

where n is the number of observations, andxiis the i-th observation.

Compound gamma

If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a

gamma distribution for the inverse-scale forms a conjugate prior. The compound distribution, which results from

integrating out the inverse-scale has a closed form solution, known as the compound gamma distribution.

Others IfX ~ Gamma(1, ), thenX has an exponential distribution with rate parameter .

IfX ~ Gamma(/2, 2), thenX is identical to 2(), the chi-squared distribution with degrees of freedom.

Conversely, if Q ~ 2() and c is a positive constant, then cQ ~ Gamma(/2, 2c).

If k is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting

time until the k-th "arrival" in a one-dimensional Poisson process with intensity 1/. If

then

IfX has a MaxwellBoltzmann distribution with parameter a, then

.

IfX ~ Gamma(k, ), then follows a generalized gamma distribution with parametersp = 2, d = 2k, and

[citation needed]

. IfX ~ Gamma(k, ) distribution, then 1/X has an inverse-gamma distribution with shape parameter k and scale

parameter using the parameterization given by inverse-gamma distribution.

IfX ~ Gamma(, ) and Y ~ Gamma(, ) are independently distributed, thenX/(X+ Y) has a beta distribution

with parameters and .

IfXi~ Gamma(

i, 1) are independently distributed, then the vector (X

1/S, ...,X

n/S), where S=X

1+ ... +X

n,

follows a Dirichlet distribution with parameters 1, ...,

n.

For large k the gamma distribution converges to Gaussian distribution with mean = k and variance 2

= k2.
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The gamma distribution is the conjugate prior for the precision of the normal distribution with known mean.

The Wishart distribution is a multivariate generalization of the gamma distribution (samples are positive-definite

matrices rather than positive real numbers).

The gamma distribution is a special case of the generalized gamma distribution, the generalized integer gamma

distribution, and the generalized inverse Gaussian distribution.

Among the discrete distributions, the negative binomial distribution is sometimes considered the discrete

analogue of the Gamma distribution.

Tweedie distributions the gamma distribution is a member of the family of Tweedie exponential dispersion

models.

Applications

The gamma distribution has been used to model the size of insurance claims[17]

and rainfalls.[18]

This means that

aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process.

The gamma distribution is also used to model errors in multi-level Poisson regression models, because the

combination of the Poisson distribution and a gamma distribution is a negative binomial distribution.

In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals. [19]

Although in practice the gamma distribution often provides a good fit, there is no underlying biophysical motivation

for using it.

In bacterial gene expression, the copy number of a constitutively expressed protein often follows the gamma

distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the

mean number of protein molecules produced by a single mRNA during its lifetime.[20]

The gamma distribution is widely used as a conjugate prior in Bayesian statistics. It is the conjugate prior for the

precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential

distribution.

Notes

[1][1] See Hogg and Craig (1978, Remark 3.3.1) for an explicit motivation

[2] http:/ /commons. wikimedia.org/wiki/File:Gamma-PDF-3D-by-k.png

[3] http:/ /commons. wikimedia.org/wiki/File:Gamma-PDF-3D-by-Theta.png

[4] http:/ /commons. wikimedia.org/wiki/File:Gamma-PDF-3D-by-x.png

[5] Papoulis, Pillai,Probability, Random Variables, and Stochastic Processes, Fourth Edition

[6] Banneheka BMSG, Ekanayake GEMUPD (2009) "A new point estimator for the median of gamma distribution". Viyodaya J Science,

14:95-103

[7][7] W.D. Penny, [www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps KL-Divergences of Normal, Gamma, Dirichlet, and Wishart

densities]

[8] Minka, Thomas P. (2002) "Estimating a Gamma distribution". http:/

/

research.microsoft.

com/

en-us/

um/

people/

minka/

papers/minka-gamma. pdf

[9] Choi, S.C.; Wette, R. (1969) "Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias", Technometrics,

11(4) 683690

[10] See Chapter 9, Section 3, pages 401428.

[11][11] Devroye (1986), p. 406.

[12] Ahrens, J. H. and Dieter, U. (1982). Generating gamma variates by a modified rejection technique. Communications of the ACM, 25, 4754.

Algorithm GD, p. 53.

[13][13] Cheng, R.C.H., and Feast, G.M. Some simple gamma variate generators. Appl. Stat. 28 (1979), 290-295.

[14][14] Marsaglia, G. The squeeze method for generating gamma variates. Comput, Math. Appl. 3 (1977), 321-325.

[15] See Chapter 9, Section 3, pages 401428.

[16] Fink, D. 1995 A Compendium of Conjugate Priors (http://www.stat.columbia. edu/~cook/movabletype/mlm/CONJINTRnew+TEX.

pdf). In progress report: Extension and enhancement of methods for setting data quality objectives. (DOE contract 95 831).

[17] p. 43, Philip J. Boland, Statistical and Probabilistic Methods in Actuarial Science, Chapman & Hall CRC 2007

[18] Aksoy, H. (2000) "Use of Gamma Distribution in Hydrological Analysis" (http://journals. tubitak.gov. tr/engineering/issues/

muh-00-24-6/muh-24-6-7-9909-13.pdf), Turk J. Engin Environ Sci, 24, 419 428.
http://journals.tubitak.gov.tr/engineering/issues/muh-00-24-6/muh-24-6-7-9909-13.pdfhttp://journals.tubitak.gov.tr/engineering/issues/muh-00-24-6/muh-24-6-7-9909-13.pdfhttp://www.stat.columbia.edu/~cook/movabletype/mlm/CONJINTRnew%2BTEX.pdfhttp://www.stat.columbia.edu/~cook/movabletype/mlm/CONJINTRnew%2BTEX.pdfhttp://research.microsoft.com/en-us/um/people/minka/papers/minka-gamma.pdfhttp://research.microsoft.com/en-us/um/people/minka/papers/minka-gamma.pdfhttp://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-x.pnghttp://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-Theta.pnghttp://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-k.pnghttp://en.wikipedia.org/w/index.php?title=Exponential_distributionhttp://en.wikipedia.org/w/index.php?title=Exponential_distributionhttp://en.wikipedia.org/w/index.php?title=Normal_distributionhttp://en.wikipedia.org/w/index.php?title=Conjugate_priorhttp://en.wikipedia.org/w/index.php?title=Protein_moleculehttp://en.wikipedia.org/w/index.php?title=Constitutively_expressedhttp://en.wikipedia.org/w/index.php?title=Copy_number_analysishttp://en.wikipedia.org/w/index.php?title=Gene_expressionhttp://en.wikipedia.org/w/index.php?title=Bacterial_geneticshttp://en.wikipedia.org/w/index.php?title=Temporal_codinghttp://en.wikipedia.org/w/index.php?title=Neurosciencehttp://en.wikipedia.org/w/index.php?title=Negative_binomial_distributionhttp://en.wikipedia.org/w/index.php?title=Poisson_distributionhttp://en.wikipedia.org/w/index.php?title=Poisson_regressionhttp://en.wikipedia.org/w/index.php?title=Gamma_processhttp://en.wikipedia.org/w/index.php?title=Insurance_policyhttp://en.wikipedia.org/w/index.php?title=Exponential_dispersion_modelhttp://en.wikipedia.org/w/index.php?title=Exponential_dispersion_modelhttp://en.wikipedia.org/w/index.php?title=Tweedie_distributionshttp://en.wikipedia.org/w/index.php?title=Negative_binomial_distributionhttp://en.wikipedia.org/w/index.php?title=Generalized_inverse_Gaussian_distributionhttp://en.wikipedia.org/w/index.php?title=Generalized_integer_gamma_distributionhttp://en.wikipedia.org/w/index.php?title=Generalized_integer_gamma_distributionhttp://en.wikipedia.org/w/index.php?title=Generalized_gamma_distributionhttp://en.wikipedia.org/w/index.php?title=Wishart_distributionhttp://en.wikipedia.org/w/index.php?title=Meanhttp://en.wikipedia.org/w/index.php?title=Normal_distributionhttp://en.wikipedia.org/w/index.php?title=Conjugate_prior


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[19] J. G. Robson and J. B. Troy, "Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat", J. Opt. Soc. Am. A 4,

23012307 (1987)

[20] N. Friedman, L. Cai and X. S. Xie (2006) "Linking stochastic dynamics to population distribution: An analytical framework of gene

expression",Phys. Rev. Lett. 97, 168302.

References

R. V. Hogg and A. T. Craig (1978)Introduction to Mathematical Statistics, 4th edition. New York: Macmillan.

(See Section 3.3.)'

P. G. Moschopoulos (1985) The distribution of the sum of independent gamma random variables, Annals of the

Institute of Statistical Mathematics, 37, 541-544

A. M. Mathai (1982) Storage capacity of a dam with gamma type inputs, Annals of the Institute of Statistical

Mathematics, 34, 591-597

External links

Hazewinkel, Michiel, ed. (2001), "Gamma-distribution" (http://www.encyclopediaofmath.org/index.

php?title=p/g043300),Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Weisstein, Eric W., " Gamma distribution (http://mathworld.wolfram.com/GammaDistribution.html)",

MathWorld.

Engineering Statistics Handbook (http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm)
http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htmhttp://en.wikipedia.org/w/index.php?title=MathWorldhttp://mathworld.wolfram.com/GammaDistribution.htmlhttp://en.wikipedia.org/w/index.php?title=Eric_W._Weissteinhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-1-55608-010-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Springer_Science%2BBusiness_Mediahttp://en.wikipedia.org/w/index.php?title=Encyclopedia_of_Mathematicshttp://www.encyclopediaofmath.org/index.php?title=p/g043300http://www.encyclopediaofmath.org/index.php?title=p/g043300


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Article Sources and Contributors 12

Article Sources and ContributorsGamma distribution Source: http://en.wikipedia.org/w/index.php?oldid=602019254 Contributors: A5, Aastrup, Abtweed98, Adam Clark, Adfernandes, Albmont, Ambarish, Amonet, Aple123,

Apocralyptic, AppliedMathematics, Arg, Asteadman, Autopilot, Baccyak4H, Barak, Bdmy, Benwing, Berland, Berzoi075, Bethb88, BeyondNormality, Bitlemon, Bo Jacoby, Bobmath, Bobo192,

Brenton, Bryan Derksen, Btyner, CanadianLinuxUser, CapitalR, Cburnett, Cerberus0, ClaudeLo, Cmghim925, Colonies Chris, Complex01, Darin, David Haslam, Dicklyon, Dlituiev, Dobromila,

Donmegapoppadoc, Dougg, DrMicro, Dshutin, Entropeneur, Entropeter, Erik144, Eug, Evan Aad, Fangz, Fnielsen, Frau K, Frobnitzem, Gaius Cornelius, Gandalf61, Gauss, Giftlite, Gjnaasaa,

Henrygb, Hgkamath, Illia Connell, Iwaterpolo, Jason Goldstick, Jirka6, Jlc46, JonathanWilliford, Jshadias, Kastchei, Lambiam, Langmore, Linas, Lovibond, LoyalSoldier, LukeSurl,

Luqmanskye, MarkSweep, Mathstat, Maththerkel, Mcld, Mebden, Melcombe, Mich8611, Michael Hardy, Mikhail Ryazanov, MisterSheik, Mpaa, MrOllie, MuesLee, Mundhenk, Narc813,

Nickfeng88, O18, PAR, PBH, Patrke, Paul Pogonyshev, Paulginz, Perspectiva8, Phil Boswell, Pichote, Plasticspork, Policron, Popnose, Qiuxing, Quietbritishjim, Qwfp, Qzxpqbp, RSchlicht,Rjwilmsi, Rlendog, Robbyjo, Robinh, Rockykumar1982, Rodionova.alenka, Samsara, Sandrobt, Schmock, Shorespirit, Smmurphy, Stephreg, Stevvers, Sun Creator, Supergrane, Talgalili, Tayste,

Tengyaow, TestUser001, Thomas stieltjes, Thric3, Tom.Reding, Tomi, Tommyjs, True rover, Umpi77, User A1, Velella, Vminin, WJVaughn3, Wavelength, Wiki me, Wiki5d, Wikid77, Wile E.

Heresiarch, Wjastle, Wqwz.wqwz1, Xuehuit, Zvika, 373 anonymous edits

Image Sources, Licenses and ContributorsImage:Gamma distribution pdf.svg Source: http://en.wikipedia.org/w/index.php?title=File:Gamma_distribution_pdf.svg License: Creative Commons Attribution-ShareAlike 3.0 Unported

Contributors: Gamma_distribution_pdf.png: MarkSweep and Cburnett derivative work: Autopilot (talk)

Image:Gamma distribution cdf.svg Source: http://en.wikipedia.org/w/index.php?title=File:Gamma_distribution_cdf.svgLicense: Creative Commons Attribution-ShareAlike 3.0 Unported

Contributors: Gamma_distribution_cdf.png: MarkSweep and Cburnett derivative work: Autopilot (talk)

Image:Gamma-PDF-3D.png Source: http://en.wikipedia.org/w/index.php?title=File:Gamma-PDF-3D.png License: Creative Commons Attribution-Sharealike 3.0 Contributors:

User:Ronhjones

Image:Gamma-KL-3D.png Source: http://en.wikipedia.org/w/index.php?title=File:Gamma-KL-3D.png License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:Ronhjones

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gamma distribution

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shape parameter

aconjugate prior distribution

mean parameter

inverse scale parameter

scale parameters