Rollout designs for lump-sum data

This work studies rollout design problems with a focus of suitable choices of rollout rate under the standard Type I and Type II error probabilities control framework. The main challenge of rollout design is that data is often observed in a lump-sum manner from a spatio-temporal point of view: (1) temporally, only the sum of data in a given sliding time window can be observed; (2) spatially, there are two subgroups for the data at each time step: control and treatment, but one can only observe the total values instead of individual values from each subgroup. We develop rollout tests of lump-sum data under both fixed-sample-size and sequential settings, subject to the constraints on Type I and Type II error probabilities. Numerical studies are conducted to validate our theoretical results.