  We consider the problem of asynchronous online testing, aimed at providing
control of the false discovery rate (FDR) during a continual stream of data
collection and testing, where each test may be a sequential test that can start
and stop at arbitrary times. This setting increasingly characterizes real-world
applications in science and industry, where teams of researchers across large
organizations may conduct tests of hypotheses in a decentralized manner. The
overlap in time and space also tends to induce dependencies among test
statistics, a challenge for classical methodology, which either assumes (overly
optimistically) independence or (overly pessimistically) arbitrary dependence
between test statistics. We present a general framework that addresses both of
these issues via a unified computational abstraction that we refer to as
"conflict sets." We show how this framework yields algorithms with formal FDR
guarantees under a more intermediate, local notion of dependence. We illustrate
our algorithms in simulations by comparing to existing algorithms for online
FDR control.
