Random clouds and an application to censoring in survival analysis

The theory of optional stopping is extended from stopping sets to general adapted random sets called ‘clouds’ and ‘anti-clouds’, and a stopping theorem is proven for set-indexed martingales. An application to set-indexed survival analysis is given when the data points are indexed by sets and censored very generally by clouds. This type of censoring corresponds to filtering of survival data on R+. A Nelson–Aalen estimator is defined, and shown to be consistent and asymptotically unbiased.