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On a theory of martingales for censoring

Benjamin R. Baer, Robert L. Strawderman

Abstract

A theory of martingales for censoring is developed. The Doob-Meyer martingale is shown to be inadequate in general, and a repaired martingale is proposed with a non-predictable centering term. Associated martingale transforms, variation processes, and covariation processes are developed based on a measure of half-predictability that generalizes predictability. The development is applied to study the Kaplan Meier estimator.

On a theory of martingales for censoring

Abstract

A theory of martingales for censoring is developed. The Doob-Meyer martingale is shown to be inadequate in general, and a repaired martingale is proposed with a non-predictable centering term. Associated martingale transforms, variation processes, and covariation processes are developed based on a measure of half-predictability that generalizes predictability. The development is applied to study the Kaplan Meier estimator.
Paper Structure (30 sections, 27 theorems, 134 equations)

This paper contains 30 sections, 27 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Observable and latent random variables
  4. Key definitions
  5. Basic Theoretical Developments
  6. Doob-Meyer martingales
  7. Identification in the latent variable model
  8. A repaired censoring martingale
  9. Comparison
  10. Martingale Transforms
  11. Martingale transforms
  12. Variation and covariation processes
  13. Alternative Development: Doob-Meyer Martingales and Predictability
  14. Applications to the Kaplan Meier Estimator
  15. Application I: Variance
  16. ...and 15 more sections

Key Result

Proposition 3.1.1

$M_T^{\sharp}$ and $M_C^{\sharp}$ are the unique Doob-Meyer $\mathcal{F}_t$-martingales.

Theorems & Definitions (54)

  • Proposition 3.1.1
  • Theorem 3.2.1
  • Proposition 3.2.1
  • Theorem 3.3.1
  • Proposition 3.3.1
  • Corollary 3.4.1
  • Lemma 4.0.1
  • Theorem 4.1.1
  • Theorem 4.2.1
  • Lemma 5.0.1
  • ...and 44 more