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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. [22]
In practice the normal distribution is often parameterized in terms of the squared scale , which corresponds to the variance of the distribution. The gamma distribution is usually parameterized in terms of a scale parameter θ {\displaystyle \theta } or its inverse.
It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution , the Weibull distribution and the gamma distribution ) are special cases of the generalized gamma, it is sometimes used ...
Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. Examples: If X is a Binomial( n , p ) random variable, and parameter p is a random variable with beta( α , β ) distribution, then X is distributed as a Beta-Binomial( α , β , n ).
Note that If (,) (Gamma distribution with scale parameter ) then / (, /) Inverse gamma distribution is a special case of type 5 Pearson distribution A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution .
The chi-squared distribution is a special case of the gamma distribution, in that (,) using the rate parameterization of the gamma distribution (or (,) using the scale parameterization of the gamma distribution) where k is an integer.
Gaussian scale mixtures: [3] [4] Compounding a normal distribution with variance distributed according to an inverse gamma distribution (or equivalently, with precision distributed as a gamma distribution) yields a non-standardized Student's t-distribution. [5]
The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known, scale parameter . [1] When the shape parameter of a Gompertz ...