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

TL;DR

This paper introduces explicit deterministic transformations, called extended one-liners, for generating Beta, Gamma, and Dirichlet distributions with shape parameters below one using elementary operations.

Contribution

It provides novel explicit transformations for these distributions with shape parameters less than one, expanding the class of distributions with simple generation methods.

Findings

Explicit transformations for Beta( a, 1-a ) with 0<a<1.

Extensions to Beta, Gamma, and Dirichlet distributions with parameters less than one.

Transformations use only elementary operations, simplifying sampling procedures.

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.