Missing at random: a stochastic process perspective

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing-data framework, we give a novel characterization of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness-at-random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration. These measurability conditions ensure the usual factorization of likelihood ratios. We illustrate how the theory can be extended easily to incorporate explanatory variables, to describe longitudinal data in continuous time, and to admit more general coarsening of observations.

Automated summary

The paper presents a natural and extensible measure-theoretic approach to handling missingness at random and characterizes observed data within the standard missing-data framework. It is demonstrated that common missingness-at-random conditions are equivalent to specific stochastic processes being adapted to a set-indexed filtration, ensuring the usual factorization of likelihood ratios. The theory is shown to easily incorporate explanatory variables, describe longitudinal data in continuous time, and allow for a more general coarsening of observations.