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The generalized gamma distribution is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
with two
shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. t ...
s (and a
scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...
). It is a generalization of the
gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distr ...
which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in
survival analysis Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysi ...
(such as the
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
, the
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...
and the
gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distr ...
) 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 In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the h ...
.


Characteristics

The generalized gamma distribution has two
shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. t ...
s, d > 0 and p > 0, and a
scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...
, a > 0. For non-negative ''x'' from a generalized gamma distribution, the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
is : f(x; a, d, p) = \frac, where \Gamma(\cdot) denotes the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...
. The
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
is : F(x; a, d, p) = \frac , where \gamma(\cdot) denotes the
lower incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...
. The
quantile function In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equ ...
can be found by noting that F(x; a, d, p) = G((x/a)^p) where G is the cumulative distribution function of the gamma distribution with parameters \alpha = d/p and \beta = 1. The quantile function is then given by inverting F using known relations about inverse of composite functions, yielding: : F^(q; a, d, p) = a \cdot \big G^(q) \big, with G^(q) being the quantile function for a gamma distribution with \alpha = d/p,\, \beta = 1.


Related distributions

* If d=p then the generalized gamma distribution becomes the
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...
. * If p=1 the generalised gamma becomes the
gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distr ...
. * If p=d=1 then it becomes the
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
. * If p=2 and d=2m then it becomes the
Nakagami distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ...
. * If p=2 and d=1 then it becomes a
half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the h ...
. Alternative parameterisations of this distribution are sometimes used; for example with the substitution ''α  =   d/p''.Johnson, N.L.; Kotz, S; Balakrishnan, N. (1994) ''Continuous Univariate Distributions, Volume 1'', 2nd Edition. Wiley. (Section 17.8.7) 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 Luigi Amoroso (26 March 1886 – 28 October 1965) was an Italian neoclassical economist influenced by Vilfredo Pareto. He provided support for and influenced the economic policy during the fascist regime. Work The microeconomical concept of the ...
who described it in 1925.


Moments

If ''X'' has a generalized gamma distribution as above, then :\operatorname(X^r)= a^r \frac .


Properties

Denote ''GG(a,d,p)'' as the generalized gamma distribution of parameters ''a'', ''d'', ''p''. Then, given c and \alpha two positive real numbers, if f \sim GG(a,d,p), then c f\sim GG(c a,d,p) and f^\alpha\sim GG\left(a^\alpha,\frac,\frac\right).


Kullback-Leibler divergence

If f_1 and f_2 are the probability density functions of two generalized gamma distributions, then their Kullback-Leibler divergence is given by : \begin D_ (f_1 \parallel f_2) & = \int_^ f_1(x; a_1, d_1, p_1) \, \ln \frac \, dx\\ & = \ln \frac + \left \frac + \ln a_1 \right (d_1 - d_2) + \frac \left( \frac \right)^ - \frac \end where \psi(\cdot) is the
digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictl ...
.C. Bauckhage (2014), Computing the Kullback-Leibler Divergence between two Generalized Gamma Distributions, .


Software implementation

In the R programming language, there are a few packages that include functions for fitting and generating generalized gamma distributions. Th
gamlss
package in R allows for fitting and generating many different distribution families includin

(family=GG). Other options in R, implemented in the package ''flexsurv'', include the function ''dgengamma'', with parameterization: \mu=\ln a + \frac, \sigma=\frac, Q=\sqrt, and in the package ''ggamma'' with parametrisation a = a, b = p, k = d/p.


See also

* Half-''t'' distribution *
Truncated normal distribution In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
*
Folded normal distribution The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
*
Rectified Gaussian distribution In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (con ...
*
Modified half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution In statistics, a normal distribution or Gaussian distribution is a type ...
* Generalized integer gamma distribution


References

{{ProbDistributions, continuous-semi-infinite Continuous distributions