Gamma distribution

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Gamma
Probability density function
Cumulative distribution function
Parameters	$k > 0\,$ shape (real) $\theta > 0\,$ scale (real)
Support	$x \in [0; \infty)\!$
Probability density function (pdf)	$x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!$
Cumulative distribution function (cdf)	$\frac{\gamma(k, x/\theta)}{\Gamma(k)}\,\!$
Mean	$k \theta\,\!$
Median	no simple closed form
Mode	$(k-1) \theta\,\!$ for $k \geq 1\,\!$
Variance	$k \theta^2\,\!$
Skewness	$\frac{2}{\sqrt{k}}\,\!$
Excess kurtosis	$\frac{6}{k}\,\!$
Entropy	$k + \ln\theta + \ln\Gamma(k) \!$ $+ (1-k)\psi(k) \!$
Moment-generating function (mgf)	$(1 - \theta\,t)^{-k}\,\!$ for $t < 1/\theta\,\!$
Characteristic function	$(1 - \theta\,i\,t)^{-k}\,\!$

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of $k$ exponentially distributed random variables, each of which has mean $θ$ .

1 Characterization
- 1.1 Probability density function
- 1.2 Cumulative distribution function
2 Properties
3 Maximum likelihood estimation
4 Generating Gamma random variables
5 Related distributions
- 5.1 Specializations
- 5.2 Others
6 References
7 See also

[edit] Characterization

[edit] Probability density function

The probability density function of the gamma distribution can be expressed in terms of the gamma function:

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

where $k > 0$ is the shape parameter and $θ > 0$ is the scale parameter of the gamma distribution. (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:

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

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

[edit] Cumulative distribution function

The cumulative distribution function can be expressed in terms of the incomplete gamma function,

$F(x;k,\theta) = \int_0^x f(u;k,\theta)\,du =\frac{\gamma(k, x/\theta)}{\Gamma(k)} \,\!$

[edit] Properties

[edit] Summation

If X_i has a Gamma(α_i, β) distribution for i = 1, 2, ..., N, then

$\left[ Y = \sum_{i=1}^N X_i \right] \sim \mathrm{Gamma} \left( \sum_{k=1}^N \alpha_i, \beta \right) \,\!$

provided all X_i are independent.

The gamma distribution exhibits infinite divisibility.

[edit] Scaling

For any t > 0 it holds that tX is distributed Gamma(k, tθ). That demonstrates that θ is a scale parameter.

[edit] Exponential family

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

[edit] Information entropy

The information entropy is given by:

$\frac{-1}{\theta^k \Gamma(k)} \int_0^{\infty} \frac{x^{k-1}}{e^{x/\theta}} \left[ (k-1)\ln x - x/\theta - k \ln\theta - \ln\Gamma(k) \right] \,dx \!$

$= -\left[ (k-1) (\ln\theta + \psi(k)) - k - k \ln\theta - \ln\Gamma(k) \right] \!$

$= k + \ln\theta + \ln\Gamma(k) + (1-k)\psi(k) \!$

where ψ(k) is the digamma function.

[edit] Kullback-Leibler divergence

The directed Kullback-Leibler divergence between Gamma(α₀, β₀) and Gamma(α, β) is given by

$\Delta(\alpha,\beta || \alpha_0, \beta_0) = \alpha \log \beta - \alpha_0 \log \beta_0 - \log \frac{ \Gamma(\alpha) }{ \Gamma(\alpha_0) } + (\alpha - \alpha_0)(\psi(\alpha) - \log \beta) - \alpha \left( 1 - \frac{\beta_0}{\beta} \right)$

[edit] Maximum likelihood estimation

The likelihood function for N iid observations $(x_1,\ldots,x_N)$ is

$L=\prod_{i=1}^N f(x_i;k,\theta)\,\!$

from which we calculate the log-likelihood function

$\ell = (k-1) \sum_{i=1}^N \ln{(x_i)} - \sum x_i/\theta - Nk\ln{(\theta)} - N\ln{\Gamma(k)}$

Finding the maximum with respect to $θ$ by taking the derivative and setting it equal to zero yields the maximum likelihood estimate 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.^{[citation needed]} 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:

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

and

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

For details, see Choi and Wette (1969).

[edit] Generating Gamma random variables

Given the scaling property above, it is enough to generate Gamma variables with β = 1 as we can later convert to any value of β 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:

$\sum_{k=1}^n {-\ln U_k} \sim \mathrm{Gamma}(n, 1),$

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(δ, 1) for 0 < δ < 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:

Let m be 1.
Generate $V 2 m - 1$ and $V 2 m$ — independent uniformly distributed on (0, 1] variables.
If $V_{2m - 1} \le v_0$ , where $v_0 = \frac e {e + \delta}$ , then go to step 4, else go to step 5.
Let $\xi_m = \left( \frac {V_{2m - 1}} {v_0} \right) ^{\frac 1 \delta}, \ \eta_m = V_{2m} \xi _m^ {\delta - 1}$ . Go to step 6.
Let $\xi_m = 1 - \ln {\frac {V_{2m - 1} - v_0} {1 - v_0}}, \ \eta_m = V_{2m} e^{-\xi_m}$ .
If $\eta_m > \xi_m^{\delta - 1} e^{-\xi_m}$ , then increment m and go to step 2.
Assume $ξ = ξ m$ to be the realization of $Gamma(δ,1)$ .

Now, to summarize,

$\theta \left( \xi - \sum _{i=1} ^{[k]} {\ln U_i} \right) \sim \mathrm{Gamma}(k, \theta),$

where [k] is the integral part of k, and ξ has been generating using the algorithm above with δ = {k} (the fractional part of k), U_k and V_l are distributed as explained above and are all independent.

[edit] Related distributions

[edit] Specializations

If $X \sim {\rm Gamma}(k=1, \theta=1/\lambda)\,$ , then X has an exponential distribution with rate parameter λ.
If $X \sim {\rm Gamma}(k=v/2, \theta=2)\,$ , then X is identical to χ²(ν), the chi-square distribution with ν degrees of freedom.
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 $X^2 \sim {\rm Gamma}(3/2, 2a^2)\,$ , then X has a Maxwell-Boltzmann distribution with parameter a.

[edit] Others

If X has a Gamma(k, θ) distribution, then 1/X has an inverse-gamma distribution with parameters k and θ^-1.
If X and Y are independently distributed Gamma(α, θ) and Gamma(β, θ) respectively, then X / (X + Y) has a beta distribution with parameters α and β.
If X_i are independently distributed Gamma(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. This holds true for any θ.

[edit] References

[edit] See also

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular

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