Risk-Controlling Predictive Sets for Time-Series Events Under Selective Observation with Finite-Sample Guarantees

Selective labels create a support failure for prediction along dependent stochastic processes: alert-triggered events are observed, whereas silent periods are usually unlabeled. We model this mechanism as predictable inclusion on a filtered probability space and show that population risk is non-identifiable when any silent region has zero labeling probability. Selective-observation weighted risk control (SOWRC) combines alert labels with randomized audits through Horvitz–Thompson losses and a martingale-mixture boundary. It provides finite-sample calibration-population control under arbitrary temporal dependence subject to predictable design choices, conditional ignorability, positivity, bounded losses, and deterministic design envelopes, together with a prospective guarantee under an externally certified deployment-drift envelope and explicit error allocation. Extensions cover anytime monitoring, multiple losses, adaptive budgets, and estimated propensities. Synthetic maintenance and financial studies, a complete-log replay on a real dependent sensor series with 100 audit-mask replications, and 4000 selection-level validation runs demonstrate support recovery and conservative probabilistic risk control on deterministic threshold grids.