Reliable simulation of extremely-truncated log-concave distributions

Inverse transform sampling is a general method to generate non-uniform-distributed random numbers, but it can be unstable when simulating extremely truncated distributions. Many famous probability distributions are log-concave; this feature is preserved under truncation, so Devroye's automatic rejection sampler is available for this task. This method is more stable than inverse transform sampling and uses a very simple envelope with an acceptance rate greater than 20%, independent of the distribution. The aim of this paper is threefold: firstly, to warn the public against incorrect simulation of truncated distributions; secondly, to motivate a more extensive use of rejection sampling to mitigate these issues; lastly, to propose Devroye's automatic sampler as a practical standard for log-concave distributions. We illustrate the proposal with simulations based on Tweedie distributions due to their relevance in regression analysis. The proposed sampler is shown to work under more extreme truncations than the inverse transform method.