With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.
With a shape parameter k and a mean parameter μ = k/β.
In each of these three forms, both parameters are positive real numbers.
The gamma distribution is the maximum entropy probability distribution for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).^{[2]}
The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution.^{[3]}
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. A random variable X that is gamma-distributed with shape α and rate β is denoted
If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion:^{[5]}
Illustration of the gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5 and 6. One can see each θ layer by itself here [2] as well as by k[3] and x. [4].
The skewness is equal to $2/{\sqrt {k}}$, it depends only on the shape parameter (k) and approaches a normal distribution when k is large (approximately when k > 10).
Unlike the mode and the mean which have readily calculable formulas based on the parameters, the median does not have an easy closed form equation. The median for this distribution is defined as the value ν such that
A formula for approximating the median for any gamma distribution, when the mean is known, has been derived based on the fact that the ratio μ/(μ − ν) is approximately a linear function of k when k ≥ 1.^{[6]} The approximation formula is
$\nu \approx \mu {\frac {3k-0.8}{3k+0.2}},$
where $\mu (=k\theta )$ is the mean.
A rigorous treatment of the problem of determining an asymptotic expansion and bounds for the median of the Gamma Distribution was handled first by Chen and Rubin, who proved
$m-{\frac {1}{3}}<\lambda (m)<m,$
where $\lambda (m)$ denotes the median of the ${\text{Gamma}}(m,1)$ distribution.^{[7]}
K.P. Choi later showed that the first five terms in the asymptotic expansion of the median are
Indeed, we know that if X is an exponential r.v. with rate λ then c X is an exponential r.v. with rate λ / c; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g., these notes, 10.4-(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale).
Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. Here β = β_{0} + 1 which are set to 1, 2, 3, 4, 5 and 6. The typical asymmetry for the KL divergence is clearly visible.
The Kullback–Leibler divergence (KL-divergence), of Gamma(α_{p}, β_{p}) ("true" distribution) from Gamma(α_{q}, β_{q}) ("approximating" distribution) is given by^{[10]}
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:
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
Generating gamma-distributed random variables[edit]
Given the scaling property above, it is enough to generate gamma variables with θ = 1 as we can later convert to any value of β with simple division.
Suppose we wish to generate random variables from Gamma(n+δ,1), where n is a non-negative integer and 0 < δ < 1. 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:
$-\sum _{k=1}^{n}{\ln U_{k}}\sim \Gamma (n,1)$
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.
Random generation of gamma variates is discussed in detail by Devroye,^{[13]}^{:401–428} noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid.^{[13]}^{:406} For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter^{[14]} modified acceptance–rejection method Algorithm GD (shape k ≥ 1), or transformation method^{[15]} when 0 < k < 1. Also see Cheng and Feast Algorithm GKM 3^{[16]} or Marsaglia's squeeze method.^{[17]}
where $\scriptstyle \lfloor {k}\rfloor$ is the integral part of k, ξ is generated via the algorithm above with δ = {k} (the fractional part of k) and the U_{k} are all independent.
While the above approach is technically correct, Devroye notes that it is linear in the value of k and in general is not a good choice. Instead he recommends using either rejection-based or table-based methods, depending on context.^{[13]}^{:401–428}
For example, Marsaglia's simple transformation-rejection method relying on a one normal and one uniform random number:^{[18]}
Setup: d=a-1/3, c=1/sqrt(9d).
Generate: v=(1+c*x)ˆ3, with x standard normal.
if v>0 and log(UNI) < 0.5*xˆ2+d-d*v+d*log(v) return d*v.
go back to step 2.
With $1\leq a=\alpha =k$ generates a gamma distributed random number in time that is approximately constant with k. The acceptance rate does depend on k, with an acceptance rate of 0.95, 0.98, and 0.99 for k=1, 2, and 4. For k<1, one can use $\gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }$ to boost k to be usable with this method.
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.^{[20]}
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.
If X ~ Gamma(1, 1/λ) (shape -scale parametrization), then X has an exponential distribution with rate parameter λ.
If X ~ Gamma(ν/2, 2), then X is identical to χ^{2}(ν), the chi-squared distribution with ν degrees of freedom. Conversely, if Q ~ χ^{2}(ν) and c is a positive constant, then cQ ~ Gamma(ν/2, 2c).
If k is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the kth "arrival" in a one-dimensional Poisson process with intensity 1/θ. If
If X ~ Gamma(k, θ), then ${\sqrt {X}}$ follows a generalized gamma distribution with parameters p = 2, d = 2k, and $a={\sqrt {\theta }}$^{[citation needed]} .
More generally, if X ~ Gamma(k,θ), then $X^{q}$ for $q>0$ follows a generalized gamma distribution with parameters p = 1/q, d = k/q, and $a=\theta ^{q}$.
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 equivalenly, ${\frac {X_{1}}{X_{2}}}\sim \beta ^{'}(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}})$
Parametrization 2: If $X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,$ are independent, then ${\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})$, or equivalently, ${\frac {X_{1}}{X_{2}}}\sim \beta ^{'}(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}})$
If X ~ Gamma(α, θ) and Y ~ Gamma(β, θ) are independently distributed, then X/(X + Y) has a beta distribution with parameters α and β.
If X_{i} ~ Gamma(α_{i}, 1) are independently distributed, then the vector (X_{1}/S, ..., X_{n}/S), where S = X_{1} + ... + X_{n}, follows a Dirichlet distribution with parameters α_{1}, ..., α_{n}.
For large k the gamma distribution converges to Gaussian distribution with mean μ = kθ and variance σ^{2} = kθ^{2}.
The Wishart distribution is a multivariate generalization of the gamma distribution (samples are positive-definite matrices rather than positive real numbers).
The gamma distribution has been used to model the size of insurance claims^{[21]} and rainfalls.^{[22]} This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process. The gamma distribution is also used to model errors in multi-level Poisson regression models, because the combination of the Poisson distribution and a gamma distribution is a negative binomial distribution.
In wireless communication, the gamma distribution is used to model the multi-path fading of signal power.
In bacterialgene expression, the copy number of a constitutively expressed protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of protein molecules produced by a single mRNA during its lifetime.^{[25]}
In genomics, the gamma distribution was applied in peak calling step (i.e. in recognition of signal) in ChIP-chip^{[26]} and ChIP-seq^{[27]} data analysis.
The gamma distribution is widely used as a conjugate prior in Bayesian statistics. 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.
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