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arxiv: 2604.11199 · v1 · submitted 2026-04-13 · 📊 stat.CO · math.PR

Recognition: unknown

Extended One-Liners for the Beta, Gamma, and Dirichlet Distributions with Shape Parameters Below One

Dylan Greaves
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:34 UTC · model grok-4.3

classification 📊 stat.CO math.PR
keywords beta distributiongamma distributiondirichlet distributionrandom variate generationuniform random variablesexact samplingshape parameter less than one
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The pith

A fixed number of uniform random variables can be transformed exactly into a Beta(a,1-a) sample for 0 < a < 1 using only elementary operations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit method to generate samples from the Beta distribution with parameters a and 1-a where a is between zero and one by applying a deterministic formula to a fixed set of independent uniform random variables. This is significant because many standard generation techniques for Beta, Gamma, and Dirichlet distributions encounter difficulties or require approximations when shape parameters fall below one. The approach avoids infinite processes or hidden approximations, relying solely on elementary operations. If correct, it enables precise simulation for these important families of distributions in statistical modeling.

Core claim

We present an explicit deterministic transformation of a fixed number of i.i.d. uniform random variables with exact Beta(a,1-a) law for 0<a<1, using only elementary operations (an extended one-liner). As corollaries, the families Beta(a,b) with min(a,b)<1, Gamma(c) with c<1, and Dirichlet(α1,…,αd) with 0<αi<1, for fixed d, also have extended one-liners.

What carries the argument

The extended one-liner: an explicit deterministic mapping from a fixed finite number of uniform[0,1] variables to the target distribution using only elementary operations.

Load-bearing premise

The proposed transformation must be exactly distributed as Beta(a,1-a) for any a in (0,1) and must rely only on a fixed finite number of uniforms together with elementary operations, with no approximations or infinite processes involved.

What would settle it

For a specific value such as a=0.25, generate a large collection of samples via the stated transformation and apply a distribution test such as the Kolmogorov-Smirnov statistic against the theoretical Beta(0.25,0.75) law; a clear mismatch would show the method is not exact.

Figures

Figures reproduced from arXiv: 2604.11199 by Dylan Greaves.

Figure 1
Figure 1. Figure 1: The function αa(p) on (0, 1) for a ∈ {0.05, 0.10, . . . , 0.95}. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

We present an explicit deterministic transformation of a fixed number of i.i.d. uniform random variables with exact Beta$(a,1-a)$ law for $0<a<1$, using only elementary operations (an ``extended one-liner'', see \cite{devroye1996oneline}). As corollaries, the families Beta$(a,b)$ with $\min(a,b)<1$, Gamma$(c)$ with $c<1$, and Dirichlet$(\alpha_1,\dots,\alpha_d)$ with $0<\alpha_i<1$, for fixed $d$, also have extended one-\liners.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper presents an explicit deterministic transformation of a fixed number of i.i.d. Uniform[0,1] random variables to a random variable with exact Beta(a,1-a) distribution for any 0<a<1, using only elementary operations. It derives corollaries providing similar extended one-liner constructions for the Beta(a,b) family when min(a,b)<1, the Gamma(c) distribution for c<1, and the Dirichlet distribution with all shape parameters less than 1 (fixed dimension d).

Significance. If the constructions hold, the work is significant for random variate generation in computational statistics. It supplies simple, exact, finite, and parameter-free methods for distributions whose standard generators often require rejection sampling, infinite series, or other complexities when shape parameters lie in (0,1). The purely constructive approach from uniform properties, without approximations or iteration counts depending on parameters, is a clear strength and directly extends the one-liner framework.

minor comments (2)
  1. The abstract and introduction should explicitly state the exact number of uniform random variables required by the Beta(a,1-a) map (and how this number depends on a, if at all) to make the 'fixed number' claim immediately verifiable.
  2. In the corollary sections deriving the Gamma and Dirichlet cases, include a brief remark on how the elementary operations compose without introducing non-elementary functions or hidden limits.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the work, recognition of its significance for random variate generation, and recommendation of minor revision. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; purely constructive derivation from uniform properties

full rationale

The manuscript claims an explicit, deterministic map from a fixed finite number of i.i.d. Uniform[0,1] variables to an exact Beta(a,1-a) random variable (0<a<1) using only elementary operations, with corollaries for Beta(a,b) when min(a,b)<1, Gamma(c) for c<1, and Dirichlet with all alpha_i<1. This is a direct constructive derivation whose validity rests on verifying that the stated map induces the target density; no parameter fitting, self-citation chains, ansatz smuggling, or renaming of known results is indicated. The derivation chain is therefore self-contained against external benchmarks and does not reduce any prediction or central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard properties of uniform random variables and the definition of the Beta distribution; no free parameters or invented entities are introduced.

axioms (1)
  • standard math i.i.d. uniform random variables on [0,1] can be transformed via elementary functions to other continuous distributions
    Core assumption underlying all random variate generation by inversion or transformation methods.

pith-pipeline@v0.9.0 · 5391 in / 1191 out tokens · 56714 ms · 2026-05-10T15:34:10.841998+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Pearson IV distribution: Random variate generation and applications

    stat.CO 2026-05 unverdicted novelty 5.0

    Uniformly fast random variate generators for the Pearson IV distribution valid over all shape parameters with applications in Bayesian statistics.

Reference graph

Works this paper leans on

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    Art B. Owen. Practical Quasi-Monte Carlo Integration. https://artowen.su.domains/mc/ practicalqmc.pdf, 2023. A Appendix A.1 Two Uniform Variant The ratioP = U 1/a 1 U 1/a 1 +U 1/(1−a) 2 has a simple closed-form CDF: FP (p) =    (1−a) ( p 1−p )a , 0<p ≤1 2, 1−a (1−p p )1−a , 1 2≤p< 1. Using the inversion method, we can generateF−1 P (U) d=P from a s...

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