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	$(k-1) \theta\text{ for }k \geq 1\,\!$
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)	$(1 - \theta\,t)^{-k}\text{ for }t < 1/\theta\,\!$
Characteristic function	$(1 - \theta\,i\,t)^{-k}\,\!$

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

[edit] Characterization

That a random variable X is gamma-distributed with scale θ and shape k is denoted

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

[edit] 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:

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

(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:

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

α

is a positive integer, then

Γ(α) = (α - 1)!.

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

[edit] Cumulative distribution function

The cumulative distribution function is the regularized gamma function, which 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 Γ(k_i, θ) distribution for i = 1, 2, ..., N, then

$\sum_{i=1}^N X_i \sim \Gamma \left( \sum_{i=1}^N k_i, \theta \right) \,\!$

provided all X_i are independent.

The gamma distribution exhibits infinite divisibility.

[edit] Scaling

For any t > 0 it holds that tX is distributed Γ(k, tθ), demonstrating 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 Γ(α₀, β₀) ('true' distribution) and Γ(α, β) ('approximating' distribution) is given by

$D_{\mathrm{KL}}(\alpha_0,\beta_0 || \alpha, \beta) = \log\left(\frac{\Gamma({\alpha})\beta_0^{\alpha}}{\Gamma(\alpha_0)\beta^{\alpha}}\right)+(\alpha_0-{\alpha})\psi(\alpha_0)+\alpha_0\frac{\beta-\beta_0}{\beta_0}$

[edit] Laplace transform

The Laplace transformation of the gamma distribution is

$F(s)=\frac{\beta^\alpha}{(s+\beta)^\alpha}.$

[edit] Parameter estimation

[edit] Maximum likelihood estimation

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

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

from which we calculate the log-likelihood function

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

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] Bayesian minimum mean-squared error

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

$P(\theta | k, x_1, ..., x_N) \propto 1/\theta \prod_{i=1}^N f(x_i;k,\theta).\,\!$

Denoting

$y \equiv \sum_{i=1}^N x_i , \qquad P(\theta | k, x_1, \dots , x_N) = C(x_i) \theta^{-N k-1} e^{-y / \theta}. \!$

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

$\int_0^{\infty} \theta^{-N k-1+m} e^{-y / \theta}\, d\theta = \int_0^{\infty} x^{N k -1 -m} e^{-x y} \, dx = y^{-(N k -m)} \Gamma(N k -m). \!$

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

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

which shows that the mean +/- standard deviation estimate of the posterior distribution for theta is

$\frac {y} {N k -1}$ +/- $\frac {y^2} {(N k-1)^2 (N k-2)}.$

[edit] Generating gamma-distributed 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 Γ(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 Γ(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 Γ(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 $V 3 m - 2$ , $V 3 m - 1$ and $V 3 m$ — independent uniformly distributed on (0, 1] variables.
If $V_{3m - 2} \le v_0$ , where $v_0 = \frac e {e + \delta}$ , then go to step 4, else go to step 5.
Let $\xi_m = V_{3m - 1}^{1 / \delta}, \ \eta_m = V_{3m} \xi _m^ {\delta - 1}$ . Go to step 6.
Let $\xi_m = 1 - \ln {V_{3m - 1}}, \ \eta_m = V_{3m} e^{-\xi_m}$ .
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 $Γ(δ,1)$

Now, to summarize,

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

where [k] is the integral part of k, and ξ has been generated 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.

The GNU Scientific Library has robust routines for sampling many distributions including the Gamma distribution.

[edit] Related distributions

[edit] Specializations

If $X \sim {\Gamma}(k=1, \theta=1/\lambda)\,$ , then X has an exponential distribution with rate parameter λ.
If $X \sim {\Gamma}(k=v/2, \theta=2)\,$ , 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 $X^2 \sim {\Gamma}(3/2, 2a^2)\,$ , then X has a Maxwell-Boltzmann distribution with parameter a.
$X \sim \mathrm{SkewLogistic}(\theta)\,$ , then $\mathrm{log}(1 + e^{-X}) \sim \Gamma (1,\theta)\,$

[edit] 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 $μ = (k - 1)θ$ and variance $σ 2 = (k - 1)θ 2$ .
The Gamma distribution is the conjugate prior for the precision of the normal distribution with known mean.

[edit] Applications

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 which is frequently modeled with a gamma distribution.^[1]

[edit] Notes

^ See Hogg and Craig Remark 3.3.1. for an explicit motivation.test

[edit] References

R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)
Eric W. Weisstein, Gamma distribution at 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

[edit] See also

Gamma distribution on wikibooks

v • d • e

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval

Beta · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Categories: Continuous distributions | Factorial and binomial topics

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