I recently ran across a new probability distribution called the “Scaled Beta2” or “SBeta2” for short in [1].

For positive argument *x* and for positive parameters *p*, *q*, and *b*, its density is

This is a heavy-tailed distribution. For large *x*, the probability density is *O*(*x*^{–q-1}), the same as a Student-*t* distribution with *q* degrees of freedom.

The authors in [1] point out several ways this distribution can be useful in application. It can approximate a number of commonly-used prior distributions while offering advantages in terms of robustness and analytical tractability.

## Related posts

- Robust prior illustration
- Probability distribution relationship diagram
- Conjugate prior relationship diagram

[1] The Scaled Beta2 Distribution as a Robust Prior for Scales. María-Eglée Pérez, Luis Raúl Pericchi, Isabel Cristina Ramírez. Available here.

Thanks a lot John

For the comment on our paper.

Hopefully this will get people away from “non informative” Inverted Gamma. Notice that the Scaled Beta 2 can be thought as an F distribution (scaled) and that is the way we call it in the generalization paper for multivariate problems. Thanks for your blog Always very informative

Luis Pericchi

University of Puerto Rico

San Juan

This appears very similar to the Generalized Pareto distribution as named and used in actuarial science for many decades. See Distribution A.2.3.1 on page 9 of . The Generalized Pareto as used in EVT (McNeil 1997 etc.) is more of a reparameteriztion of the classic (called ballasted in actuarial literature) Pareto where theta/q is less correlated with alpha.

It seems that putting the link in brackets has hidden it. The PDF is https://www.casact.org/admissions/syllabus/MASI_Tables.pdf

https://www.casact.org/admissions/syllabus/MASI_Tables.pdf

The un-scaled version is the distribution for odds, p/(1-p). This is a homework problem I give my stats majors; transform the beta distribution of p into the the distribution for the odds.

I recently had an application of the un-scaled version, for a problem in which I was interested in calculating the mean: https://www.landonlehman.com/post/2021-02-25-bayesian-estimation-of-multinomial-parameter-ratios-a-derivation/.

I didn’t know the distribution had a name.