Missing at random: a stochastic process perspective

Area of Science:

• Statistics
• Probability Theory
• Data Science

Background:

• Missing data is a common challenge in statistical analysis.
• Existing frameworks for missingness at random (MAR) have limitations.
• A rigorous theoretical foundation is needed for advanced missing data methods.

Purpose of the Study:

• To develop a natural and extensible measure-theoretic treatment of missingness at random.
• To provide a novel characterization of observed data within the MAR framework.
• To demonstrate the equivalence of MAR conditions to stochastic process adaptivity.

Main Methods:

• Utilizing measure theory and stochastic processes.
• Characterizing observed data as a stopping-set sigma algebra.
• Defining missingness-at-random conditions via adaptedness to set-indexed filtrations.

Main Results:

• The observed data is characterized as a stopping-set sigma algebra.
• Missingness-at-random conditions are shown to be equivalent to specific stochastic process adaptivity requirements.
• Measurability conditions ensure the factorization of likelihood ratios.

Conclusions:

• The proposed measure-theoretic framework offers a robust and extensible treatment of missing data.
• The theory naturally extends to incorporate explanatory variables and continuous-time longitudinal data.
• This approach accommodates more general forms of observation coarsening.