Scaled Beta2 distribution

Posted on by John

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

\frac{\Gamma(p+q)}{b\,\Gamma(p) \, \Gamma(q)}\, \frac{\left(\frac{x}{b}\right)^{p-1}}{\left(\left(\frac{x}{b}\right) + 1 \right)^{p+q}}

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

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

5 thoughts on “Scaled Beta2 distribution”

  1. 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

  2. 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.

  4. 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.

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