## ratio-of-uniforms [#4]

**P**ossibly the last post on random number generation by Kinderman and Monahan’s (1977) *ratio-of-uniform method*. After fiddling with the Gamma(a,1) distribution when a<1 for a while, I indeed figured out a way to produce a bounded set with this method: considering an arbitrary cdf Φ with corresponding pdf φ, the uniform distribution on the set Λ of (u,v)’s in **R⁺**x**X** such that

0≤u≤Φο**ƒ**[φοΦ⁻¹(u)v]

induces the distribution with density proportional to **ƒ** on φοΦ⁻¹(U)V. This set Λ has a boundary that is parameterised as

u=Φο**ƒ**(x), v=1/φο**ƒ**(x), x∈Χ

which remains bounded in u since Φ is a cdf and in v if φ has fat enough tails. At both 0 and ∞. When **ƒ** is the Gamma(a,1) density this can be achieved if φ behaves like log(x)² near zero and like a inverse power at infinity. Without getting into all the gory details, closed form density φ and cdf Φ can be constructed for all a’s, as shown for a=½ by the boundaries in u and v (yellow) below

which leads to a bounded associated set Λ

At this stage, I remain uncertain of the relevance of such derivations, if only because the set A thus derived is ill-suited for uniform draws proposed on the enclosing square box. And also because a Gamma(a,1) simulation can rather simply be derived from a Gamma(a+1,1) simulation. But, who knows?!, there may be alternative usages of this representation, such as innovative slice samplers. Which means the ratio-of-uniform method may reappear on the ‘Og one of those days…

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