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The false coverage rate (FCR) is the expected ratio of number of constructedconfidence intervals (CIs) that fail to cover their respective parameters tothe total number of constructed CIs. Procedures for FCR control exist in theoffline setting, but none so far have been designed with the online setting inmind. In the online setting, there is an infinite sequence of fixed unknownparameters $\theta_t$ ordered by time. At each step, we see independent datathat is informative about $\theta_t$, and must immediately make a decisionwhether to report a CI for $\theta_t$ or not. If $\theta_t$ is selected forcoverage, the task is to determine how to construct a CI for $\theta_t$ suchthat $\text{FCR} \leq \alpha$ for any $T\in \mathbb{N}$. A straightforwardsolution is to construct at each step a $(1-\alpha)$ level conditional CI. Inthis paper, we present a novel solution to the problem inspired by online falsediscovery rate (FDR) algorithms, which only requires the statistician to beable to construct a marginal CI at any given level. Apart from the fact thatmarginal CIs are usually simpler to construct than conditional ones, themarginal procedure has an important qualitative advantage over the conditionalsolution, namely, it allows selection to be determined by the candidate CIitself. We take advantage of this to offer solutions to some online problemswhich have not been addressed before. For example, we show that our general CIprocedure can be used to devise online sign-classification procedures thatcontrol the false sign rate (FSR). In terms of power and length of theconstructed CIs, we demonstrate that the two approaches have complementarystrengths and weaknesses using simulations. Last, all of our methodologyapplies equally well to online FCR control for prediction intervals, havingparticular implications for assumption-free selective conformal inference.

Online Control of the False Coverage Rate and False Sign Rate