A modified adaptive accept–reject algorithm for univariate densities with bounded support

Abstract

The need to simulate from a univariate density arises in several settings, particularly in Bayesian analysis. An especially efficient algorithm which can be used to sample from a univariate density, f _X , is the adaptive accept–reject algorithm. To implement the adaptive accept–reject algorithm, the user has to envelope T ∘ f _X , where T is some transformation such that the density g(x) ∝ T ⁻¹ (α+β x) is easy to sample from. Successfully enveloping T ∘ f _X , however, requires that the user identify the number and location of T ∘ f _X ’s inflection points. This is not always a trivial task. In this paper, we propose an adaptive accept–reject algorithm which relieves the user of precisely identifying the location of T ∘ f _X ’s inflection points. This new algorithm is shown to be efficient and can be used to sample from any density such that its support is bounded and its log is three-times differentiable.

Keywords:

• log-concave
• T-concave
• T-convex
• inflection points
• bayesian analysis