# Ratio of Uniforms

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Part of the Statistics and Computing book series (SCO)

## Abstract

This chapter provides a detailed description of the so-called ratio-of-uniforms (RoU) methods. The RoU and the generalized RoU (GRoU) techniques were introduced in Kinderman and Monahan (ACM Trans Math Softw 3(3):257–260, 1977); Wakefield et al. (Stat Comput 1(2):129–133, 1991) as bivariate transformations of the bidimensional region $$\mathcal {A}_0$$ below the target pdf p o (x) ∝ p(x). To be specific, the RoU techniques can be seen as a transformation of a bidimensional uniform random variable, defined over $$\mathcal {A}_0$$, into another two-dimensional random variable defined over an alternative set $$\mathcal {A}$$. RoU schemes also convert samples uniformly distributed on $$\mathcal {A}$$ into samples with density p o (x) ∝ p(x) (which is equivalent to draw uniformly from $$\mathcal {A}_0$$). Therefore, RoU methods are useful when drawing uniformly from the region $$\mathcal {A}$$ is comparatively simpler than drawing from p o (x) itself (i.e., simpler than drawing uniformly from $$\mathcal {A}_0$$). In general, RoU algorithms are applied in combination with the rejection sampling principle and they turn out especially advantageous when $$\mathcal {A}$$ is bounded. In this chapter, we present first the basic theory underlying RoU methods, and then study in depth the connections with other sampling techniques. Several extensions, as well as different variants and point of views, are discussed.

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1. 1.

In the fundamental theorem of simulation of Sect. 2.4.3, the target pdf p(x) is assumed unnormalized in general, so that multiplying p(x) for a positive constant does not change the results of the theorem.

2. 2.

Recall that the inequalities depend on the sign of the first derivative of p(x) (i.e., whether p(x) is increasing or decreasing).

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