Stratified sampling

The recently arXived paper of Goldstein, Rinott and Scarsini studies the impact of refining a partition on the precision of a stratified maximising/integration Monte Carlo approach. Quite naturally, if the partition gets improved, simulating points in each set of the partition can only improve the quality of the approximation, whether the problem is in maximising or in integrating. However, the authors include an interesting (formal) counterexample where the stratification leads to a higher L₁ (if not L₂) error. (And they include extensions of more classical results to cases when the function is observed with errors or contains missing data.) The difficulty I have with stratification in practice is that it is really difficult to come up with a partition which is relevant for the problem at hand and whose partition weights are known exactly… Reading this nice mathematical paper also led me to ponder the possibility of doing unbiased maximisation, while wondering if stratification was eventually compelling for maximisation, since only one set in the partition is of interest. Eliminating sets from the simulation would thus be leading to higher efficiency, if this was feasible.