Gamma

Template:Probability distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. It has a scale parameter θ and a shape parameter k. If k is an integer then the distribution represents the sum of k independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of θ ⁻¹) .

The gamma distribution is frequently a probability model for waiting times; for instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution.<ref>See Hogg and Craig Remark 3.3.1. for an explicit motivation.test</ref> Gamma distributions were fitted to rainfall amounts from different storms, and differences in amounts from seeded and unseeded storms were reflected in differences in estimated k and $\theta$ parameters <ref>Template:Citation</ref>

Characterization

A random variable X that is gamma-distributed with scale θ and shape k is denoted

$X\sim \Gamma (k,\theta ){\text{ or }}X\sim {\textrm {Gamma}}(k,\theta ).\,$

Probability density function

The probability density function of the gamma distribution can be expressed in terms of the gamma function parameterized in terms of a shape parameter k and scale parameter θ. Both k and θ will be positive values.

The equation defining the probability density function of a gamma-distributed random variable x is

$f(x;k,\theta )=x^{k-1}{\frac {e^{-x/\theta }}{\theta ^{k}\,\Gamma (k)}}{\text{ for }}x>0{\text{ and }}k,\theta >0.\,$

(This parameterization is used in the infobox and the plots.)

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter:

$g(x;\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta {x}}{\text{ for }}x>0.\,$

If α is a positive integer, then

$\Gamma (\alpha )=(\alpha -1)!.\,$

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

File:Gamma-PDF-3D.png

Illustration of the Gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5 and 6. One can see each θ layer by itself here [1] as well as by k [2] and x. [3].

Cumulative distribution function

The cumulative distribution function is the regularized gamma function:

$F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma (k,x/\theta )}{\Gamma (k)}}\,$

where $\gamma (k,x/\theta )$ 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)<ref>Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, Fourth Edition</ref>:

$F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {(x/\theta )^{i}}{i!}}e^{-x/\theta }.$

Properties

Summation

If X_i has a Γ(k_i, θ) distribution for i = 1, 2, ..., N, then

$\sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)\,\!$

provided all X_i' are independent.

The gamma distribution exhibits infinite divisibility.

Scaling

If

$X\,\sim \,\mathrm {Gamma} (k,\theta )$

then for any α > 0,

$\alpha X\,\sim \,\mathrm {Gamma} (k,\alpha \theta ).$

Exponential family

The Gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/θ, and natural statistics X and ln (X).

Information entropy

The information entropy is given by

${\begin{aligned}&{}\qquad {\frac {-1}{\theta ^{k}\Gamma (k)}}\int _{0}^{\infty }{\frac {x^{k-1}}{e^{x/\theta }}}\left[(k-1)\ln x-x/\theta -k\ln \theta -\ln \Gamma (k)\right]\,dx\\[8pt]&=-\left[(k-1)(\ln \theta +\psi (k))-k-k\ln \theta -\ln \Gamma (k)\right]\\[8pt]&=k+\ln \theta +\ln \Gamma (k)+(1-k)\psi (k)\end{aligned}}$

where ψ(k) is the digamma function.

One can also show that (if we use the shape parameter k and the inverse scale parameter β),

$\mathbb {E} [\ln(x)]=\psi (k)-\ln(\beta ).\,$

Or alternately, using the scale parameter θ,

$\mathbb {E} [\ln(x)]=\psi (k)+\ln(\theta ).\,$

File:Gamma-KL-3D.png

Illustration of the Kullback–Leibler (KL) divergence for two Gamma PDF's. Here β = β₀ + 1 which are set to 1, 2, 3, 4, 5 and 6. The typical asymmetry for the KL divergence is clearly visible.

Kullback–Leibler divergence

The directed Kullback–Leibler divergence between Γ(θ₀, β₀) ('true' distribution) and Γ(θ, β) ('approximating' distribution), for shape parameter θ and inverse scale parameter β is given by

$D_{\mathrm {KL} }(\theta _{0},\beta _{0}||\theta ,\beta )=\log \left({\frac {\Gamma ({\theta })\beta _{0}^{\theta _{0}}}{\Gamma (\theta _{0})\beta ^{\theta }}}\right)+(\theta _{0}-\theta )(\psi (\theta _{0})-\log \beta _{0})+\theta _{0}{\frac {\beta -\beta _{0}}{\beta _{0}}}$

Laplace transform

The Laplace transform of the gamma PDF is

$F(s)=\left(1+\theta s\right)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.$

Parameter estimation

Maximum likelihood estimation

The likelihood function for N iid observations (x₁, ..., x_N) is

$L(k,\theta )=\prod _{i=1}^{N}f(x_{i};k,\theta )\,\!$

from which we calculate the log-likelihood function

$\ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln {(x_{i})}-\sum _{i=1}^{N}x_{i}/\theta -Nk\ln {(\theta )}-N\ln {\Gamma (k)}.$

Finding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the maximum likelihood estimator of the θ parameter:

${\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}.\,\!$

Substituting this into the log-likelihood function gives

$\ell =(k-1)\sum _{i=1}^{N}\ln {(x_{i})}-Nk-Nk\ln {\left({\frac {\sum x_{i}}{kN}}\right)}-N\ln {(\Gamma (k))}.\,\!$

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

$\ln {(k)}-\psi (k)=\ln {\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)}-{\frac {1}{N}}\sum _{i=1}^{N}\ln {(x_{i})}\,\!$

where

$\psi (k)={\frac {\Gamma '(k)}{\Gamma (k)}}\!$

is the digamma function.

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 the method of moments, or using the approximation

$\ln(k)-\psi (k)\approx {\frac {1}{k}}\left({\frac {1}{2}}+{\frac {1}{12k+2}}\right).\,\!$

If we let

$s=\ln {\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)}-{\frac {1}{N}}\sum _{i=1}^{N}\ln {(x_{i})},\,\!$

then k is approximately

$k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}$

which is within 1.5% of the correct value.Template:Fact An explicit form for the Newton-Raphson update of this initial guess is given by Choi and Wette (1969) as the following expression:

$k\leftarrow k-{\frac {\ln k-\psi \left(k\right)-s}{1/k-\psi \;'\left(k\right)}}$

where $\psi \;'\left(\cdot \right)$ denotes the trigamma function (the derivative of the digamma function).

The digamma and trigamma functions can be difficult to calculate with high precision. However, approximations known to be good to several significant figures can be computed using the following approximation formulae:

$\psi \left(k\right)={\begin{cases}\ln(k)-(1+(1-(1/10-1/(21k^{2}))/k^{2})/(6k))/(2k),\quad k\geq 8\\\psi \left(k+1\right)-1/k,\quad k<8\end{cases}}$

and

$\psi \;'\left(k\right)={\begin{cases}(1+(1+(1-(1/5-1/(7k^{2}))/k^{2})/(3k))/(2k))/k,\quad k\geq 8,\\\psi \;'\left(k+1\right)+1/k^{2},\quad k<8.\end{cases}}$

For details, see Choi and Wette (1969).

Bayesian minimum mean-squared error

With known k and unknown $\theta$, the posterior PDF for theta (using the standard scale-invariant prior for $\theta$) is

$P(\theta |k,x_{1},...,x_{N})\propto 1/\theta \prod _{i=1}^{N}f(x_{i};k,\theta ).\,\!$

Denoting

$y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta |k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }.\!$

Integration over θ can be carried out using a change of variables, revealing that 1/θ is gamma-distributed with parameters $\scriptstyle \alpha =Nk,\ \ \beta =y$.

$\int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m).\!$

The moments can be computed by taking the ratio (m by m = 0)

$E(x^{m})={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m},\!$

which shows that the mean ± standard deviation estimate of the posterior distribution for theta is

${\frac {y}{Nk-1}}\pm {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}.$

Generating gamma-distributed random variables

Given the scaling property above, it is enough to generate gamma variables with $\theta =1$ as we can later convert to any value of $\beta$ 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 "α-addition" property of gamma distribution, we expand this result:

$\sum _{k=1}^{n}{-\ln U_{k}}\sim \Gamma (n,1),$

where $U_{k}$ are all uniformly distributed on $(0,1]$ and independent.

All that is left now is to generate a variable distributed as $\Gamma (\delta ,1)$ for $0<\delta <1$ and apply the "α-addition" property once more. This is the most difficult part.

We provide an algorithm without proof. It is an instance of the acceptance-rejection method:

1. Let $m$ be 1.
2. Generate $V_{3m-2}$, $V_{3m-1}$ and $V_{3m}$ as independent uniformly distributed on $(0,1]$ variables.
3. If $V_{3m-2}\leq v_{0}$, where $v_{0}={\frac {e}{e+\delta }}$, then go to step 4, else go to step 5.
4. Let $\xi _{m}=V_{3m-1}^{1/\delta },\ \eta _{m}=V_{3m}\xi _{m}^{\delta -1}$. Go to step 6.
5. Let $\xi _{m}=1-\ln {V_{3m-1}},\ \eta _{m}=V_{3m}e^{-\xi _{m}}$.
6. If $\eta _{m}>\xi _{m}^{\delta -1}e^{-\xi _{m}}$, then increment $m$ and go to step 2.
7. Assume $\xi =\xi _{m}$ to be the realization of $\Gamma (\delta ,1)$

Now, to summarize,

$\theta \left(\xi -\sum _{i=1}^{\lfloor {k}\rfloor }{\ln U_{i}}\right)\sim \Gamma (k,\theta ),$

where $\lfloor {k}\rfloor$ is the integral part of $k$, and $\xi$ has been generated using the algorithm above with $\delta =\{k\}$ (the fractional part of $k$), $U_{k}$ and $V_{l}$ are distributed as explained above and are all independent.

Related distributions

Specializations

• If $X\sim {\Gamma }(k=1,\theta =1/\lambda )\,$, then X has an exponential distribution with rate parameter λ.
• If $X\sim {\Gamma }(k=\nu /2,\theta =2)\,$, then X is identical to χ²(ν), the chi-square distribution with ν degrees of freedom. Conversely, if $Q\sim {\chi }^{2}(\nu )\,$ and c is a positive constant, then $c\cdot Q\sim {\Gamma }(k=\nu /2,\theta =2c)\,$.
• 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/θ.
• If $X^{2}\sim {\Gamma }(3/2,2a^{2})\,$, then X has a Maxwell-Boltzmann distribution with parameter a.
• $X\sim \mathrm {SkewLogistic} (\theta )\,$, then $\mathrm {log} (1+e^{-X})\sim \Gamma (1,\theta )\,$, i.e. an exponential distribution: see skew-logistic distribution.

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 σ, and inverse gamma with known shape parameter.

The Gamma distribution's conjugate prior is <ref name="fink">Fink, D. 1995 A Compendium of Conjugate Priors. In progress report: Extension and enhancement of methods for setting data quality objectives. (DOE contract 95‑831).</ref>:

$p(k,\theta |p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}}$

Where Z is the normalizing constant, which has no closed form solution. The posterior distribution can be found by updating the parameters as follows.

$p'=p\prod _{i}x_{i}$ $q'=q+\sum _{i}x_{i}$ $r'=r+n$,$s'=s+n$

Where $n$ is the number of observations, and $x_{i}$ is the $i^{th}$ observation.

Others

• If X has a Γ(k, θ) distribution, then 1/X has an inverse-gamma distribution with parameters k and θ^-1.
• If X and Y are independently distributed Γ(α, θ) and Γ(β, θ) respectively, then X / (X + Y) has a beta distribution with parameters α and β.
• If X_i are independently distributed Γ(α_i,θ) respectively, then the vector (X₁ / S, ..., X_n / S), where S = X₁ + ... + X_n, follows a Dirichlet distribution with parameters α₁, ..., α_n.
• For large k the gamma distribution converges to Gaussian distribution with mean $\mu =k\theta$ and variance $\sigma ^{2}=k\theta ^{2}$.
• 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.
• Among the discrete distributions, the negative binomial distribution is sometimes considered the discrete analogue of the Gamma distribution

Applications

Template:Expand-section The gamma distribution has been used to model the size of insurance claims and rainfalls. This means aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process.

See also

• Gamma process
• Lukacs's proportion-sum independence theorem

Notes

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References

• R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)
• Template:MathWorld
• Engineering Statistics Handbook
• S. C. Choi and R. Wette. (1969) Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias, Technometrics, 11(4) 683–690