Time-uniform confidence bands for the CDF under nonstationarity

Estimation of the complete distribution of a random variable is a useful primitive for both manual and automated decision making. This problem has received extensive attention in the i.i.d. setting, but the arbitrary data dependent setting remains largely unaddressed. Consistent with known impossibility results, we present computationally felicitous time-uniform and value-uniform bounds on the CDF of the running averaged conditional distribution of a real-valued random variable which are always valid and sometimes trivial, along with an instance-dependent convergence guarantee. The importance-weighted extension is appropriate for estimating complete counterfactual distributions of rewards given controlled experimentation data exhaust, e.g., from an A/B test or a contextual bandit.