Gamma distribution

:then ::\Pr(X x) = \Pr(Y - If X has a Maxwell–Boltzmann distribution with parameter a, then ::X^2 \sim \Gamma{\left(\tfrac{3}{2}, 2a^2\right)}. - Y \sim N(\mu = \alpha \lambda, \sigma^2 = \alpha \lambda^2) is a normal distribution as Y = \lim_{\alpha \to \infty} X where *X* ~ Gamma(*α*, *λ*). -- - If *X* ~ Gamma(*α*, *θ*), then \exp X follows a log-gamma distribution. - If *X* ~ Gamma(*α*, *θ*), then \log X follows an exponential-gamma (abbreviated exp-gamma) distribution. It is sometimes incorrectly referred to as the log-gamma distribution. Formulas for its mean and variance are in the section #Logarithmic expectation and variance. - If *X* ~ Gamma(*α*, *θ*), then \sqrt{X} follows a generalized gamma distribution with parameters , , and a = \sqrt{\theta}. - More generally, if *X* ~ Gamma(*α*,*θ*), then X^q for q 0 follows a generalized gamma distribution with parameters , , and a = \theta^q. - If *X* ~ Gamma(*α*, *θ*) with shape α and scale θ, then 1/*X* ~ Inv-Gamma(*α*, *θ*−1) (see Inverse-gamma distribution for derivation). - Parametrization 1: If X_k \sim \Gamma(\alpha_k,\theta_k)\, are independent, then \frac{\alpha_2\theta_2 X_1}{\alpha_1\theta_1 X_2} \sim \mathrm{F}(2\alpha_1, 2\alpha_2), or equivalently, \frac{X_1}{X_2} \sim \lambda'\left(\alpha_1, \alpha_2, 1, \frac{\theta_1}{\theta_2}\right) - Parametrization 2: If X_k \sim \Gamma(\alpha_k,\lambda_k)\, are independent, then \frac{\alpha_2\lambda_1 X_1}{\alpha_1\lambda_2 X_2} \sim \mathrm{F}(2\alpha_1, 2\alpha_2), or equivalently, \frac{X_1}{X_2} \sim \lambda'\left(\alpha_1, \alpha_2, 1, \frac{\lambda_2}{\lambda_1}\right) - If *X* ~ Gamma(*α*, *θ*) and *Y* ~ Gamma(*λ*, *θ*) are independently distributed, then *X*/(*X* + *Y*) has a beta distribution with parameters α and λ, and *X*/(*X* + *Y*) is independent of *X* + *Y*, which is Gamma(*α* + *λ*, *θ*)-distributed. - If X_n \sim \text{Beta}(\alpha,n\lambda)\, and Y_n = nX_n, then Y_n converges in distribution to \text{Gamma}(\alpha,\lambda) defined under parametrization 2. - If *X**i* ~ Gamma(*α**i*, 1) are independently distributed, then the vector (*X*1/*S*, ..., *X**n*/*S*), where , follows a Dirichlet distribution with parameters *α*1, ..., *α**n*. - For large α the gamma distribution converges to normal distribution with mean and variance . - The gamma distribution is the conjugate prior for the precision of the normal distribution with known mean. - The matrix gamma distribution and the Wishart distribution are multivariate generalizations 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, the generalized integer gamma distribution, and the generalized inverse Gaussian distribution. - Among the discrete distributions, the negative binomial distribution is sometimes considered the discrete analog of the gamma distribution. - Tweedie distributions – the gamma distribution is a member of the family of Tweedie exponential dispersion models. - Modified Half-normal distribution – the Gamma distribution is a member of the family of Modified half-normal distribution. The corresponding density is f(x\mid \alpha, \lambda, \gamma)= \frac{2\lambda^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\lambda x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\lambda}}\right)}}, where \Psi(\alpha,z) = {}_1\Psi_1{\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right)} denotes the Fox–Wright Psi function. - For the shape-scale parameterization x|\theta \sim \Gamma(\alpha,\theta), if the scale parameter \theta \sim IG(b,1) where IG denotes the Inverse-gamma distribution, then the marginal distribution x \sim \lambda'(\alpha,b) where \lambda' denotes the Beta prime distribution. ### Compound gamma If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The compound distribution, which results from integrating out the inverse scale, has a closed-form solution known as the compound gamma distribution. If, instead, the shape parameter is known but the mean is unknown, with the prior of the mean being given by another gamma distribution, then it results in K-distribution. ## Statistical inference ### Parameter estimation #### Maximum likelihood estimation The likelihood function for N iid observations (*x*1, ..., *x**N*) is L(\alpha, \theta) = \prod_{i=1}^N f(x_i;\alpha,\theta) from which we calculate the log-likelihood function \ell(\alpha, \theta) = (\alpha - 1) \sum_{i=1}^N \ln x_i - \sum_{i=1}^N \frac{x_i} \theta - N\alpha\ln \theta - N\ln \Gamma(\alpha) 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, which equals the sample mean \bar{x} divided by the shape parameter α: \hat{\theta} = \frac{1}{\alpha N}\sum_{i=1}^N x_i = \frac{\bar{x}}{\alpha} Substituting this into the log-likelihood function gives \ell(\alpha) = (\alpha-1)\sum_{i=1}^N \ln x_i -N\alpha - N\alpha\ln \frac{\sum_i x_i}{\alpha N} - N\ln \Gamma(\alpha) We need at least two samples: N\ge2, because for N=1, the function \ell(\alpha) increases without bounds as \alpha\to\infty. For \alpha0, it can be verified that \ell(\alpha) is strictly concave, by using inequality properties of the polygamma function. Finding the maximum with respect to α by taking the derivative and setting it equal to zero yields \begin{align} \ln \alpha - \psi(\alpha) &= \ln\left(\frac{1}{N}\sum_{i=1}^N x_i\right) - \frac 1 N \sum_{i=1}^N \ln x_i \\[1ex] &= \ln \bar{x} - \overline{\ln x} \end{align} where ψ is the digamma function and \overline{\ln x} is the sample mean of ln *x*. There is no closed-form solution for α. 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 \alpha - \psi(\alpha) \approx \frac{1}{2\alpha}\left(1 + \frac{1}{6\alpha + 1} \right) If we let \begin{align} s &= \ln \left(\frac 1 N \sum_{i=1}^N x_i\right) - \frac 1 N \sum_{i=1}^N \ln x_i \\[1ex] &= \ln \bar{x} - \overline{\ln x} \end{align} then α is approximately k \approx \frac{3 - s + \sqrt{\left(s - 3\right)^2 + 24s}}{12s} which is within 1.5% of the correct value. An explicit form for the Newton–Raphson update of this initial guess is: \alpha \leftarrow \alpha - \frac{ \ln \alpha - \psi(k) - s }{ \frac 1 \alpha - \psi\prime(\alpha) }. At the maximum-likelihood estimate (\hat \alpha,\hat\theta), the expected values for x and \ln x agree with the empirical averages: \begin{align} \hat \alpha\hat\theta &= \bar x &&\text{and} & \psi(\hat \alpha)+\ln \hat\theta &= \overline{\ln x}. \end{align} ##### Caveat for small shape parameter For data, (x_1,\ldots,x_N), that is represented in a floating point format that underflows to 0 for values smaller than \varepsilon, the logarithms that are needed for the maximum-likelihood estimate will cause failure if there are any underflows. If we assume the data was generated by a gamma distribution with cdf F(x;\alpha,\theta), then the probability that there is at least one underflow is: \Pr(\text{underflow}) = 1-(1-F(\varepsilon;\alpha,\theta))^N This probability will approach 1 for small α and large N. For example, at \alpha=10^{-2}, N=10^4 and \varepsilon=2.25\times10^{-308}, \Pr(\text{underflow})\approx 0.9998. A workaround is to instead have the data in logarithmic format. In order to test an implementation of a maximum-likelihood estimator that takes logarithmic data as input, it is useful to be able to generate non-underflowing logarithms of random gamma variates, when \alpha. Following the implementation in `scipy.stats.loggamma`, this can be done as follows: sample Y\sim\text{Gamma}(\alpha+1,\theta) and U\sim\text{Uniform} independently. Then the required logarithmic sample is Z=\ln(Y)+\ln(U)/\alpha, so that \exp(Z)\sim\text{Gamma}(k,\theta). #### Closed-form estimators There exist consistent closed-form estimators of α and θ that are derived from the likelihood of the generalized gamma distribution. The estimate for the shape α is \hat{\alpha} = \frac{N \sum\limits_{i=1}^N x_i}{N \sum\limits_{i=1}^N x_i \ln x_i - \sum\limits_{i=1}^N x_i \sum\limits_{i=1}^N \ln x_i} and the estimate for the scale θ is \hat{\theta} = \frac{1}{N^2} \left(N \sum_{i=1}^N x_i \ln x_i - \sum_{i=1}^N x_i \sum_{i=1}^N \ln x_i\right) Using the sample mean of x, the sample mean of ln *x*, and the sample mean of the product *x*·ln *x* simplifies the expressions to: \hat{\alpha} = \frac{\bar{x}}{\hat{\theta}} \hat{\theta} = \overline{x\ln x} - \bar{x} \overline{\ln x}. If the rate parameterization is used, the estimate of \hat{\lambda} = 1/\hat{\theta}. These estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators. They have however similar efficiency as the maximum likelihood estimators. Although these estimators are consistent, they have a small bias. A bias-corrected variant of the estimator for the scale θ is \tilde{\theta} = \frac{N}{N - 1} \hat{\theta} A bias correction for the shape parameter α is given as \tilde{\alpha} = \hat{\alpha} - \frac{1}{N} \left(3 \hat{\alpha} - \frac{2}{3} \left(\frac{\hat{\alpha}}{1 + \hat{\alpha}}\right) - \frac{4}{5} \frac{\hat{\alpha}}{(1 + \hat{\alpha})^2} \right) #### Bayesian minimum mean squared error With known α and unknown θ, the posterior density function for theta (using the standard scale-invariant prior for θ) is \Pr(\theta \mid \alpha, x_1, \dots, x_N) \propto \frac 1 \theta \prod_{i=1}^N f(x_i; \alpha, \theta) Denoting y \equiv \sum_{i=1}^Nx_i , \qquad \Pr(\theta \mid \alpha, x_1, \dots, x_N) = C(x_i) \theta^{-N \alpha-1} e^{-y/\theta} where the C (integration) constant does not depend on θ. The form of the posterior density reveals that 1 / *θ* is gamma-distributed with shape parameter *Nα* + 2 and rate parameter y. Integration with respect to θ can be carried out using a change of variables to find the integration constant \begin{align} \int_0^\infty \theta^{-N\alpha - 1 + m} e^{-y/\theta}\, d\theta &= \int_0^\infty x^{N\alpha - 1 - m} e^{-xy} \, dx \\ &= y^{-(N\alpha - m)} \Gamma(N\alpha - m) \! \end{align} The moments can be computed by taking the ratio (m by ) \operatorname{E} [x^m] = \frac {\Gamma (N\alpha - m)} {\Gamma(N\alpha)} y^m which shows that the mean ± standard deviation estimate of the posterior distribution for θ is \frac y {N\alpha - 1} \pm \sqrt{\frac {y^2} {\left(N\alpha - 1\right)^2 (N\alpha - 2)}}. ### Bayesian inference #### 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, inverse gamma with known shape parameter, and Gompertz with known scale parameter. The gamma distribution's conjugate prior is: p(\alpha,\theta \mid p, q, r, s) = \frac{1}{Z} \frac{p^{\alpha-1} e^{-\theta^{-1} q}}{\Gamma(\alpha)^r \theta^{\alpha s}}, where Z is the normalizing constant with no closed-form solution. The posterior distribution can be found by updating the parameters as follows: \begin{align} p' &= p\prod\nolimits_i x_i,\\ q' &= q + \sum\nolimits_i x_i,\\ r' &= r + n,\\ s' &= s + n, \end{align} where n is the number of observations, and *x**i* is the i-th observation from the gamma distribution. ## Occurrence and applications Consider a sequence of events, with the waiting time for each event being an exponential distribution with rate λ. Then the waiting time for the n-th event to occur is the gamma distribution with integer shape \alpha = n. This construction of the gamma distribution allows it to model a wide variety of phenomena where several sub-events, each taking time with exponential distribution, must happen in sequence for a major event to occur. Examples include the waiting time of cell-division events, number of compensatory mutations for a given mutation, waiting time until a repair is necessary for a hydraulic system, and so on. In biophysics, the dwell time between steps of a molecular motor like ATP synthase is nearly exponential at constant ATP concentration, revealing that each step of the motor takes a single ATP hydrolysis. If there were n ATP hydrolysis events, then it would be a gamma distribution with degree n. The gamma distribution has been used to model the size of insurance claims and rainfalls. This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process – much like the exponential distribution generates a Poisson process. The gamma distribution is also used to model errors in multi-level Poisson regression models because a mixture of Poisson distributions with gamma-distributed rates has a known closed form distribution, called negative binomial. In wireless communication, the gamma distribution is used to model the multi-path fading of signal power; see also Rayleigh distribution and Rician distribution. In oncology, the age distribution of cancer incidence often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of driver events and the time interval between them. In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals. In bacterial gene expression where protein production can occur in bursts, the copy number of a given protein often follows the gamma distribution, where the shape and scale parameters are, respectively, the mean number of bursts per cell cycle and the mean number of protein molecules produced per burst. In genomics, the gamma distribution was applied in peak calling step (i.e., in recognition of signal) in ChIP-chip and ChIP-seq data analysis. In Bayesian statistics, the gamma distribution is widely used as a conjugate prior. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential distribution. In phylogenetics, the gamma distribution is the most commonly used approach to model among-sites rate variation when maximum likelihood, Bayesian, or distance matrix methods are used to estimate phylogenetic trees. Phylogenetic analyzes that use the gamma distribution to model rate variation estimate a single parameter from the data because they limit consideration to distributions where . This parameterization means that the mean of this distribution is 1 and the variance is 1/*α*. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution. ## Random variate generation Given the scaling property above, it is enough to generate gamma variables with , as we can later convert to any value of λ with a simple division. Suppose we wish to generate random variables from Gamma(*n* + *δ*, 1), where n is a non-negative integer and {{math|0 -\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(*δ*, 1) for {{math|0 Random generation of gamma variates is discussed in detail by Devroye, noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid. For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter modified acceptance-rejection method Algorithm GD (shape *α* ≥ 1), or transformation method when {{math|0 The following is a version of the Ahrens-Dieter acceptance–rejection method: 1. Generate U, V and W as iid uniform (0, 1] variates. 1. If U\le\frac e {e+\delta} then \xi=V^{1/\delta} and \eta=W\xi^{\delta-1}. Otherwise, \xi=1-\ln V and \eta=We^{-\xi}. 1. If \eta\xi^{\delta-1}e^{-\xi} then go to step 1. 1. ξ is distributed as Γ(*δ*, 1). A summary of this is \theta \left( \xi - \sum_{i=1}^{\lfloor \alpha \rfloor} \ln U_i \right) \sim \Gamma (\alpha, \theta) where \scriptstyle \lfloor \alpha \rfloor is the integer part of α, ξ is generated via the algorithm above with (the fractional part of α) and the *U**k* are all independent. While the above approach is technically correct, Devroye notes that it is linear in the value of α and generally is not a good choice. Instead, he recommends using either rejection-based or table-based methods, depending on context. For example, Marsaglia's simple transformation-rejection method relying on one normal variate X and one uniform variate U: 1. Set d = a - \frac13 and c = \frac1{\sqrt{9d}}. 1. Set v=(1+cX)^3. 1. If v 0 and \ln U return dv, else go back to step 2. With 1 \le a = \alpha generates a gamma distributed random number in time that is approximately constant with α. The acceptance rate does depend on α, with an acceptance rate of 0.95, 0.98, and 0.99 for *α* = 1, 2, and 4. 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