Separability in optimal allocation

The optimal allocation for stratification, parameterized by the respective sampling strategy to use in each stratum, is derived directly from the notion of efficiency. Especially with simulation, there are often opportunities to maximize efficiency (myopically) within each stratum. To maximize efficiency globally, first maximize the efficiency of the sampling strategy for each stratum separately and then use the optimal allocation given these respective maximizers. Given any other allocation, maximizing the efficiency of the sampling strategy in each stratum separately does not give the highest efficiency attainable with that allocation except in degenerate cases. Given a class C of deterministic rounding strategies, the rounding of the (continuous) optimal allocation over C, which maximizes efficiency, cannot be improved by a strategy that randomizes over C.