Siqi Shao, R. A. Serota
arXiv 7 Oct 2026 · Econometrics
arXiv:2610.10476 · PDF · Extracted main text
We introduce a mean-reverting stochastic differential equation with a three-component stochastic term and show that it generates a hierarchy of steady-state (stationary) distributions. At the top level, the hierarchy is described by a modified-Beta distribution, while one- and two-parameter reductions produce compact-support, power-law-tailed, and exponential-type limiting families within a single stochastic framework. We then construct two generalized extensions of this hierarchy. In the first, the power transformation is applied directly at the level of the stochastic differential equation; in the second, the same transformation is applied only after the stationary modified-Beta hierarchy has been obtained. While these two procedures agree on important lower branches and limiting cases they generally differ at the top level. The generalized hierarchy is therefore not unique: nonlinear transformation and stationary-state reduction do not commute. For both routes, we derive the probability density and cumulative distribution functions, express their parameters in terms of the underlying stochastic dynamics, clarify the relations among their limiting cases, and compare the resulting families with the traditional Generalized Beta framework.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Liu, J. and Serota, R. A (2023) Rethinking Generalized Beta family of distributions self | 1.000 | 9 | 3 | 100% |
| 2 | NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov | 0.874 | 6 | 2 | 100% |
| 3 | McDonald, James B and Xu, Yexiao J (1995) A generlazition of the beta distribution with applications | 0.811 | 4 | 2 | 100% |
| 4 | Dashti Moghaddam, M and Serota, RA (2021) Combined Mutiplicative-Heston Model for Stochastic Volatility | 0.737 | 3 | 2 | 100% |
| 5 | Alexander, Carol and Cordeiro, Gauss M and Ortega Edwin M M and Sara… (2012) Generalized beta-generated distributions | 0.405 | 1 | 1 | 100% |
| 6 | Alzaatrech, Ayman and Lee, Carl and Famoye, Felix (2013) A new method for generating families of continuous distributions | 0.405 | 1 | 1 | 100% |
| 7 | (2018) Econometrics and Income Inequality | 0.405 | 1 | 1 | 100% |
| 8 | Bouchaud, Jean-Philippe and Mézard, Marc (2000) Wealth condensation in a simple model of economy | 0.405 | 1 | 1 | 100% |
| 9 | (2013) Econophysics of Income and Wealth Distributions | 0.405 | 1 | 1 | 100% |
| 10 | (2008) Modeling Income Distributions and Lorenz Curves | 0.405 | 1 | 1 | 100% |
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