Parameters a > 0 , d > 0 , p > 0 {\displaystyle a>0,d>0,p>0} Support x ∈ ( 0 , ∞ ) {\displaystyle x\;\in \;(0,\,\infty )} PDF p / a d Γ ( d / p ) x d − 1 e − ( x / a ) p {\displaystyle {\frac {p/a^{d}}{\Gamma (d/p)}}x^{d-1}e^{-(x/a)^{p}}} CDF γ ( d / p , ( x / a ) p ) Γ ( d / p ) {\displaystyle {\frac {\gamma (d/p,(x/a)^{p})}{\Gamma (d/p)}}} Mean a Γ ( ( d + 1 ) / p ) Γ ( d / p ) {\displaystyle a{\frac {\Gamma ((d+1)/p)}{\Gamma (d/p)}}} Mode a ( d − 1 p ) 1 p , f o r d > 1 {\displaystyle a\left({\frac {d-1}{p}}\right)^{\frac {1}{p}},\mathrm {for} \;d>1} | ||
The generalized gamma distribution is a continuous probability distribution with three parameters. It is a generalization of the two-parameter gamma distribution. 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.
Contents
Characteristics
The generalized gamma has three parameters:
where
The cumulative distribution function is
where
If
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.
Moments
If X has a generalized gamma distribution as above, then
Kullback-Leibler divergence
If
where
Software implementation
In R implemented in the package flexsurv, function dgengamma, with different parametrisation: