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Gamma distribution

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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 exponentially distributed random variables, each of which has parameter \frac{1}{\theta}.

CharacterizationEdit

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).

Probability density functionEdit

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\,\, \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 \alpha = k and an inverse scale parameter \beta = 1/\theta, called a rate parameter:

 g(x;\alpha,\beta) = x^{\alpha-1}  \frac{\beta^{\alpha} \, e^{-\beta\,x} }{\Gamma(\alpha)}  \ \mathrm{for}\ x > 0 \,\!.
If \alpha is a positive integer, then  {\Gamma(\alpha)}=(\alpha - 1)!

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

Cumulative distribution functionEdit

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)} \,\!

PropertiesEdit

SummationEdit

If Xi has a Γ(αi, β) distribution for i = 1, 2, ..., N, then


\sum_{i=1}^N X_i
\sim
\Gamma  \left( \sum_{i=1}^N \alpha_i, \beta \right) \,\!

provided all Xi are independent.

The gamma distribution exhibits infinite divisibility.

ScalingEdit

For any t > 0 it holds that tX is distributed Γ(ktθ), demonstrating that θ is a scale parameter.

Exponential familyEdit

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

Information entropyEdit

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.

Kullback–Leibler divergenceEdit

The directed Kullback–Leibler divergence between Γ(α0, β0) ('true' distribution) and Γ(α, β) ('approximating' distribution) is given by


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

Laplace transform Edit

The Laplace transformation of the gamma distribution is


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

Parameter estimation Edit

Maximum likelihood estimation Edit

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 \theta 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).

Bayesian minimum mean-squared error Edit

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^{-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)}.

Generating gamma-distributed random variables Edit

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 Uk 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:

  1. Let m be 1.
  2. Generate V_{3m - 2}, V_{3m - 1} and V_{3m} — independent uniformly distributed on (0, 1] variables.
  3. If V_{3m - 2} \le 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} ^{[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), Uk and Vl are distributed as explained above and are all independent.

The GNU Scientific Library (which has ports for Visual Studio) has robust routines for sampling many distributions including the Gamma distribution.

Related distributionsEdit

Specializations Edit

Others Edit

  • 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 Xi are independently distributed Γ(αi,θ) respectively, then the vector (X1 / S, ..., Xn / S), where S = X1 + ... + Xn, follows a Dirichlet distribution with parameters α1, ..., αn.

References Edit

See also Edit

Template:ProbDistributions

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