Generalized Gamma Distribution in Lifetime Models It seems to me that a version of generalized gamma distribution is often used in lifetime models. For example, survival analysis uses this distribution.
Is there some intuition as to why this distribution fits well to study these models? Is it simply a discovery from looking at the empirical distribution of different studies that involve birth and death?
For example, when learning Poisson distribution and random variable, students learn about an arrival process of bus at a bus stop. It is intuitive.
Is there something similar to using the generalized gamma distribution?
 A: I can think of a few reasons:

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*it is popular because it's a very general distribution with many familiar distributions as special cases. For a $\text{GeneralizedGamma}(\mu, \sigma, \lambda)$,


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*When $\lambda = 1$ and $\sigma = 1$, then the distribution is Exponential.

*When $\lambda = 1$ then the distribution is Weibull.

*When $\sigma = \lambda$ then the distribution is Gamma.

*When $\lambda = 0$ then the distribution is Log-Normal.

*When $\lambda = -1$ then the distribution is Inverse-Weibull.

*When $\sigma = -\lambda$ then the distribution is Inverse-Gamma.

Because of this, the modeller doesn't need to force their data to fit into a probably-incorrect distribution. It also allows for model refutation. Ex: if I decided a NHST that the data is lognormal, and I fit to a GG model, then if $\lambda$ is significantly far from 0, I can reject the null.


*Its flexibility means it can capture more exotic distributions, which makes prediction more accurate, at only the cost of a few additional parameters.


*It's part of the accelerated lifetime survival family.
Unfortunately, I doubt there's a "first principles" derivation of the GG model.
