Missing at random: a stochastic process perspective

TL;DR

This paper provides a rigorous, measure-theoretic framework for understanding missing at random in statistical data, linking it to stochastic processes and filtrations to clarify conditions for ignorability.

Contribution

It introduces a novel measure-theoretic characterization of missing at random using stopping-set sigma algebras and stochastic process adaptation.

Findings

Equivalent conditions for missing at random using stochastic process adaptation

Clarification of fixed versus random elements in missing data

Framework ensures likelihood factorization for ignorability

Abstract

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing data framework, we give a novel characterisation 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 of the complete data: measurability conditions that suffice to ensure the likelihood factorisation necessary for ignorability. Our rigorous statement of the missing at random conditions also clarifies a common confusion: what is fixed, and what is random?