  Online testing procedures aim to control the extent of false discoveries over
a sequence of hypothesis tests, allowing for the possibility that early-stage
test results influence the choice of hypotheses to be tested in later stages.
Typically, online methods assume that a permanent decision regarding the
current test (reject or not reject) must be made before advancing to the next
test. We instead assume that each hypothesis requires an immediate preliminary
decision, but also allows us to update that decision until a preset deadline.
Roughly speaking, this lets us apply a Benjamini-Hochberg-type procedure over a
moving window of hypotheses, where the threshold parameters for upcoming tests
can be determined based on preliminary results. Our method controls the false
discovery rate (FDR) at every stage of testing, as well as at adaptively chosen
stopping times. These results apply even under arbitrary p-value dependency
structures.
