🤖 AI Summary
This work addresses the challenge of efficiently and exactly sampling from Beta, Gamma, and Dirichlet distributions when their shape parameters are less than one—a regime where existing methods often require iterative or approximate procedures. The authors propose a novel approach based on explicit deterministic transformations that generates exact samples using only a fixed number of independent uniform random variables and elementary arithmetic operations. This method yields concise, non-iterative, and approximation-free “extended one-liners” for Beta(a,b) with min(a,b)<1, Gamma(c) with c<1, and Dirichlet(α₁,…,α_d) with 0<α_i<1. By eliminating the need for rejection sampling or numerical inversion, the scheme preserves mathematical exactness while significantly improving computational efficiency.
📝 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$(α_1,\dots,α_d)$ with $0<α_i<1$, for fixed $d$, also have extended one-\liners.