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.

Any information or helpful advice would be much appreciated.

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.


2 Answers 2


I am not sure about "better known" part of your question, but take a look at [1], namely equation (1) defining the K4D tool by its quantile function. A few pages later, the author lists several special cases of parameter values for which K4D contains other distributions, including the uniform, exponential and Gumbel distributions. The author then appears to answer your question by saying that "The generalization of the Pareto distribution proposed by Stoppa (1990) is obtained for these values of the parameters. Please take a look in the original material.

[1] http://www-c.eco.unibs.it/~stateap/vol4-nspec1/05Tarsitano.pdf


Actually, this response is nearly perfect for my purposes, because, unlike the Stoppa, a right-Kappa and several related distribution has already been fully implemented in R, including L-moments, trimmed L-moments (TL), and right-censored versions -- perfect for my purposes.

Thanks! andrewH


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