Gamma distribution
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Template:Probability distribution In probability theory and statistics, the gamma distribution is a continuous probability distribution. For integer values of the parameter k it is also known as the Erlang distribution.
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Probability density function
The probability density function of the gamma distribution can be expressed in terms of the gamma function:
- <math> f(x|k,\theta) = x^{k-1} \frac{e^{-x/\theta}}{\theta^k \, \Gamma(k)}
\ \mathrm{for}\ x > 0 \,\!<math>
where <math>k > 0<math> is the shape parameter and <math>\theta > 0<math> is the scale parameter of the gamma distribution. (NOTE: this parameterization is what is used in the infobox and the plots.)
Alternatively, the gamma distribution can be parameterized in terms of a shape parameter <math>\alpha = k<math> and an inverse scale parameter <math>\beta = 1/\theta<math>, called a rate parameter:
- <math> g(x|k,\lambda) = x^{\alpha-1} \frac{\beta^{\alpha} \, e^{-\beta\,x} }{\Gamma(\alpha)} \ \mathrm{for}\ x > 0 \,\!<math>
Both are common because they are more convenient to use in certain fields with different parameterizations.
Properties
The cumulative distribution function can be expressed in terms of the incomplete gamma function,
- <math> F(x|k,\theta) = \int_0^x f(u|k,\theta)\,du
= \frac{\gamma(k, x/\theta)}{\Gamma(k)} \,\!<math>
The information entropy is given by:
<math>S=k\theta+(1-k)\ln(\theta)+\ln(\Gamma(k))+(1-k)\psi(k)\,<math>
where <math>\psi(k)<math> is the polygamma function.
If <math>X_i \sim \mathrm{Gamma}(\alpha_i, \beta)<math> for <math>i=1, 2, \cdots, N<math> and <math>\bar{\alpha} = \sum_{k=1}^N \alpha_i<math> then
- <math>
\left[ Y = \sum_{i=1}^N X_i \right] \sim \mathrm{Gamma} \left( \bar{\alpha}, \beta \right) <math>
provided all <math>X_i<math> are independent. The gamma distribution exhibits infinite divisibility.
If <math>X \sim \operatorname {Gamma} (\alpha, \beta)<math>, then <math>\frac X \beta \sim \operatorname {Gamma} (\alpha, 1)<math>. Or, more generally, for any <math>t > 0<math> it holds that <math>tX \sim \operatorname {Gamma} (\alpha, t \beta)<math>. That is the meaning of β (or θ) being the scale parameter.
Parameter estimation
The likelihood function is
- <math>L=\prod_{i=1}^N f(x_i|k,\theta)<math>
from which we calculate the log-likelihood function
- <math>\ell=(k-1)\sum_{i=1}^N\ln(x_i)-\sum x_i/\theta-Nk\ln(\theta)-N\ln\Gamma(k))<math>
Finding the maximum with respect to <math>\theta<math> by taking the derivative an setting it equal to zero yields the maximum likelihood estimate of the <math>\theta<math> parameter:
- <math>\theta=\frac{1}{kN}\sum_{i=1}^N x_i<math>
Generating gamma random variables
Given the scaling property above, it is enough to generate gamma variables with <math>\beta = 1<math> as we can later convert to any value of β with simple division.
Using the fact that if <math>X \sim \operatorname {Gamma} (1, 1)<math>, then also <math>X \sim \operatorname {Exponential} (1)<math>, and the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then <math>-\ln U \sim \operatorname {Gamma} (1, 1)<math>. Now, using the "α-addition" property of gamma distribution, we expand this result:
- <math>\sum _{k=1} ^n {-\ln U_k} \sim \operatorname {Gamma} (n, 1)<math>,
where <math>U_k<math> are all uniformly distributed on (0, 1] and independent.
All that is left now is to generate a variable distributed as <math>\operatorname {Gamma} (\delta, 1)<math> for <math>0 < \delta < 1<math> and apply the "α-addition" property once more. This is the most difficult part, however.
We provide an algorithm without proof. It is an instance of the acceptance-rejection method:
- Let m be 1.
- Generate <math>V_{2m - 1}<math> and <math>V_{2m}<math> — independent uniformly distributed on (0, 1] variables.
- If <math>V_{2m - 1} \le v_0<math>, where <math>v_0 = \frac e {e + \delta}<math>, then go to step 4, else go to step 5.
- Let <math>\xi_m = \left( \frac {V_{2m - 1}} {v_0} \right) ^{\frac 1 \delta}, \ \eta_m = V_{2m} \xi _m^ {\delta - 1}<math>. Go to step 6.
- Let <math>\xi_m = 1 - \ln {\frac {V_{2m - 1} - v_0} {1 - v_0}}, \ \eta_m = V_{2m} e^{-\xi_m}<math>.
- If <math>\eta_m > x^{\delta - 1} e^{-x}<math> then increment m and go to step 2.
- Assume <math>\xi = \xi_m<math> to be the realization of <math>\operatorname {Gamma} (\delta, 1)<math>.
Now, to summarize,
- <math>\frac 1 \beta \left( \xi - \sum _{k=1} ^{[\alpha]} {\ln U_k} \right) \sim \operatorname {Gamma} (\alpha, \beta)<math>,
where <math>[\alpha]<math> is the integral part of α, ξ has been generating using the algorithm above with <math>\delta = \{\alpha\}<math> (the fractional part of α), <math>U_k<math> and <math>V_l<math> are distributed as explained above and are all independent.
Related distributions
- <math>X \sim \mathrm{Exponential}(\theta)<math> is an exponential distribution if <math>X \sim \mathrm{Gamma}(1, \theta)<math>.
- <math>Y \sim \mathrm{Gamma}(N, \theta)<math> is a gamma distribution if <math>Y = X_1 + \cdots + X_N<math> and if the <math>X_i \sim \mathrm{Exponential}(\theta)<math> are all independent and share the same parameter <math>\theta<math>.
- <math>X \sim \chi^2(\nu)<math> is a chi-square distribution if <math>X \sim \mathrm{Gamma}(k=\nu/2, \theta = 2)<math>.
- If <math>k<math> is an integer, the gamma distribution is an Erlang distribution (so named in honor of A. K. Erlang) and is the probability distribution of the waiting time until the <math>k^{th}<math> "arrival" in a one-dimensional Poisson process with intensity <math>1/\theta<math>.
- <math>X \sim \mathrm{Gamma}(k, \theta)<math> then <math>Y \sim \mathrm{InvGamma}(k, \theta^{-1})<math> if <math>Y = 1/X<math>, where <math>\mathrm{InvGamma}<math> is the inverse-gamma distribution.
- <math>Y = X_1/(X_1+X_2) \sim \mathrm{Beta}<math> is a beta distribution if <math>X_1 \sim \mathrm{Gamma}<math> and <math>X_2 \sim \mathrm{Gamma}<math> and are also independent.
- <math>Y \sim \mathrm{Maxwell}(\beta)<math> is a Maxwell-Boltzmann distribution if <math>X \sim \mathrm{Gamma}(\alpha = 3/2, \beta)<math>.
- <math>Y \sim N(\mu = \alpha \beta, \sigma^2 = \alpha \beta^2)<math> is a normal distribution as <math>Y = \lim_{\alpha \to \infty} X<math> where <math>X \sim \mathrm{Gamma}(\alpha, \beta)<math>.
References
- R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)