Generalized gamma distribution

The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). 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 to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution. The generalized gamma distribution has two shape parameters, and , and a scale parameter, . For non-negative x from a generalized gamma distribution, the probability density function is where denotes the gamma function. The cumulative distribution function is where denotes the lower incomplete gamma function, and denotes the regularized lower incomplete gamma function. 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 using known relations about inverse of composite functions, yielding: with being the quantile function for a gamma distribution with . If then the generalized gamma distribution becomes the Weibull distribution. If the generalised gamma becomes the gamma distribution. If then it becomes the exponential distribution. If and then it becomes the Nakagami distribution. If and then it becomes a half-normal distribution. Alternative parameterisations of this distribution are sometimes used; for example with the substitution α = d/p. In addition, a shift parameter can be added, so the domain of x starts at some value other than zero. If the restrictions on the signs of a, d and p are also lifted (but α = d/p remains positive), this gives a distribution called the Amoroso distribution, after the Italian mathematician and economist Luigi Amoroso who described it in 1925.