Generalized gamma distribution

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Generalized gamma
Parameters a>0, d>0, p>0
Support x \;\in\; (0,\, \infty)
pdf \frac{p/a^d}{\Gamma(d/p)} x^{d-1}e^{-(x/a)^p}
CDF \frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)}
Mean a \frac{\Gamma((d+1)/p)}{\Gamma(d/p)}
Mode a \left(\frac{d-1}{p}\right)^{\frac{1}{p}}, \mathrm{for}\; d>1
Variance a^2\left(\frac{\Gamma((d+2)/p)}{\Gamma(d/p)} - \left(\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}\right)^2\right)
Entropy \ln \frac{a \Gamma(d/p)}{p} + \frac{d}{p} + \left(\frac{1}{a}-\frac{d}{p}\right)\psi\left(\frac{d}{p}\right)

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 Weibull distribution and the log-normal 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.[1]

Characteristics[edit]

The generalized gamma has three parameters: a > 0, d > 0, and p > 0. For non-negative x, the probability density function of the generalized gamma is[2]


f(x; a, d, p) = \frac{(p/a^d) x^{d-1} e^{-(x/a)^p}}{\Gamma(d/p)},

where \Gamma(\cdot) denotes the gamma function.

The cumulative distribution function is


F(x; a, d, p) = \frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)} ,

where \gamma(\cdot) denotes the lower incomplete gamma function.

If d=p then the generalized gamma distribution becomes the Weibull distribution. Alternatively, if p=1 the generalised gamma becomes the gamma distribution.

Alternative parameterisations of this distribution are sometimes used; for example with the substitution α  =   d/p.[3] In addition, a shift parameter can be added, so the domain of x starts at some value other than zero.[3] 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.[4]

Moments[edit]

If X has a generalized gamma distribution as above, then[3]

\operatorname{E}(X^r)= a^r \frac{\Gamma (\frac{d+r}{p})}{\Gamma( \frac{d}{p})} .

Kullback-Leibler divergence[edit]

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{align}
D_{KL} (f_1 \parallel f_2) 
& = \int_{0}^{\infty} f_1(x; a_1, d_1, p_1) \, \ln \frac{f_1(x; a_1, d_1, p_1)}{f_2(x; a_2, d_2, p_2)} \, dx\\
& = \ln \frac{p_1 \, a_2^{d_2} \, \Gamma\left(d_2 / p_2\right)}{p_2 \, a_1^{d_1} \, \Gamma\left(d_1 /p_1\right)} 
    + \left[ \frac{\psi\left( d_1 / p_1 \right)}{p_1} + \ln a_1 \right]  (d_1 - d_2) 
    + \frac{\Gamma\bigl((d_1+p_2) / p_1 \bigr)}{\Gamma\left(d_1 / p_1\right)} \left( \frac{a_1}{a_2} \right)^{p_2} 
    - \frac{d_1}{p_1}
\end{align}

where \psi(\cdot) is the digamma function.[5]

See also[edit]

References[edit]

  1. ^ Box-Steffensmeier, Janet M.; Jones, Bradford S. (2004) Event History Modeling: A Guide for Social Scientists. Cambridge University Press. ISBN 0-521-54673-7 (pp. 41-43)
  2. ^ Stacy, E.W. (1962). "A Generalization of the Gamma Distribution." Annals of Mathematical Statistics 33(3): 1187-1192. JSTOR 2237889
  3. ^ a b c Johnson, N.L.; Kotz, S; Balakrishnan, N. (1994) Continuous Univariate Distributions, Volume 1, 2nd Edition. Wiley. ISBN 0-471-58495-9 (Section 17.8.7)
  4. ^ Gavin E. Crooks (2010), The Amoroso Distribution, Technical Note, Lawrence Berkeley National Laboratory.
  5. ^ C. Bauckhage (2014), Computing the Kullback-Leibler Divergence between two Generalized Gamma Distributions, arXiv:1401.6853.