Monotonicity and stratification

In utilizing the technique of stratification, the user needs to first partition/stratify the sample space; the next task is to determine how to allocate samples to strata. How to best perform the second task is well understood and analyzed and there are effective and generic recipes for sample allocation. Performing the first task, on the other hand, is generally left to the user who has limited guidelines at her/his disposal. We review explicit and implicit stratification approaches considered in the literature and discuss their relevance to simulation studies. We then discuss the different ways in which monotonicity plays a role in optimal stratification.