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Missing at random: a stochastic process perspective.

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Summary
This summary is machine-generated.

This study introduces a measure-theoretic approach to missing data, defining observed data as a stopping-set sigma algebra. This framework clarifies missingness-at-random conditions and extends to complex data scenarios.

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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.