# Is the Stoppa distribution a special case of any better-known, more general distribution?

The Stoppa distribution is a 3-parameter distribution that generalizes the Pareto distribution, adding a second shape parameter but no location term. The CDF is

$$F(x) = \left[1-\left(\frac{x}{x_0}\right)^{-α}\right]^θ;\quad 0 < x_0 \leq x$$

Definition from Kleiber & Kotz (2003). This distribution is becoming increasingly important in the study of income distribution at the high end.

Is there any better-known distribution for which the Stoppa is a special case, given some restriction on the parameters of the more general distribution? And if so, what is the distribution and the restriction?

I suspect it is a special case of the five-parameter distribution that McDonald & Xu (1995) calls the Generalized Beta, Crooks (2012) calls the generalized beta prime, and some other authors call the Feller-Pareto (though all these names have also been used for the four-parameter distribution McDonald calls the generalized beta of the second kind). But I have not yet been able to find a set of restrictions that gets me to the Stoppa, so this may be wrong.

And I have not been able to identify it with any member of the Pearson, Burr, Dagum, etc. distributional families, though I could easily miss it if it is just parameterized a little differently.

Crooks (2012). Survey of Simple, Continuous, Univariate Probability Distributions.

Kleiber & Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences.

McDonald & Xu (1995). A Generalization of the Beta Distribution with Applications.