  The false coverage rate (FCR) is the expected ratio of number of constructed
confidence intervals (CIs) that fail to cover their respective parameters to
the total number of constructed CIs. Procedures for FCR control exist in the
offline setting, but none so far have been designed with the online setting in
mind. In the online setting, there is an infinite sequence of fixed unknown
parameters $\theta_t$ ordered by time. At each step, we see independent data
that is informative about $\theta_t$, and must immediately make a decision
whether to report a CI for $\theta_t$ or not. If $\theta_t$ is selected for
coverage, the task is to determine how to construct a CI for $\theta_t$ such
that $\text{FCR} \leq \alpha$ for any $T\in \mathbb{N}$. A straightforward
solution is to construct at each step a $(1-\alpha)$ level conditional CI. In
this paper, we present a novel solution to the problem inspired by online false
discovery rate (FDR) algorithms, which only requires the statistician to be
able to construct a marginal CI at any given level. Apart from the fact that
marginal CIs are usually simpler to construct than conditional ones, the
marginal procedure has an important qualitative advantage over the conditional
solution, namely, it allows selection to be determined by the candidate CI
itself. We take advantage of this to offer solutions to some online problems
which have not been addressed before. For example, we show that our general CI
procedure can be used to devise online sign-classification procedures that
control the false sign rate (FSR). In terms of power and length of the
constructed CIs, we demonstrate that the two approaches have complementary
strengths and weaknesses using simulations. Last, all of our methodology
applies equally well to online FCR control for prediction intervals, having
particular implications for assumption-free selective conformal inference.
