Majumdar–Ghosh Model

Template:Probability distribution 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.

Characteristics

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

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

where $\Gamma$ 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 $\gamma$ 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. 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

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

See also

• Generalized integer gamma distribution

References

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