Search results
Results from the WOW.Com Content Network
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. For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)
The closely related inverse-gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution. If α is a positive integer, then the distribution represents an Erlang distribution; i.e., the sum of α independent exponentially distributed random variables, each of which has a mean of θ.
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters .
In mathematics, the multivariate gamma function Γ p is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution. [1] It has two equivalent definitions.
i.e., the inverse-gamma distribution, where () is the ordinary Gamma function. The Inverse Wishart distribution is a special case of the inverse matrix gamma distribution when the shape parameter = and the scale parameter =. Another generalization has been termed the generalized inverse Wishart distribution, .
A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart distribution
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding:
Usually, the inverse gamma function refers to the principal branch with domain on the real interval [, +) and image on the real interval [, +), where = … [2] is the minimum value of the gamma function on the positive real axis and = = … [3] is the location of that minimum.