TMA4275 Lifetime analysis

  • Introduction to the course
    • Time axis, censoring and truncation
    • Survival function and hazard rate
    • Counting processes
    • Modelling of event history data
  • Discrete time martingales
    • Discrete time martingale
    • Variation processes
    • Stopping times and transformations
    • Doob decomposition
  • Continuous time martingales
    • Martingale
    • Stochastic integrals
    • Doob-Meyer decomposition
    • Poisson process
    • Counting processes
    • Stochastic integrals for counting process martingales
  • The Nelson-Aalen estimator
    • Nelson-Aalen estimator
    • Handling of ties for the Nelson-Aalen estimator
    • Gaussian martingales and martingale convergence theorem
    • Large sample properties for the Nelson-Aalen estimator
  • The Kaplan-Meier estimator
    • Kaplan-Meier estimator
    • Large sample properties for the Kaplan-Meier estimator
    • Handling of ties for the Kaplan-Meier estimator
    • Estimation of median and mean survival times
    • Non-parametric tests
  • Kaplan-Meier for a Markov chain
    • Kaplan-Meier for a Markov chain: The general case
    • Competing risks
    • Illness-death model
  • Relative risk regression
    • Regression models and relative risk regression
    • Partial likelihood for relative risk regression
    • Large scale properties of betahat
    • Estimation of cumulative hazard and survival probabilities
    • Stratified relative risk model
    • Model checking in Cox regression
  • Additive regression models
    • Vector-valued counting processes, martingales and stochastic integrals
    • Estimation in the additive hazard model
    • Martingales tests for additive regression models
  • Parametric counting processes
    • Parametric counting process models: In introduction
    • Likelihood for censored survival times
    • Likelihood for counting process models
    • Maximum likelihood estimator and related tests
    • Parametric regression models
  • Frailty models
    • The odd effects of frailty
    • Multivariate frailty models
    • Shared frailty models
    • Empirical Bayes estimate of individual frailty
  • TMA4275 Lifetime analysis
  • Discrete time martingales
  • Stopping times and transformations
  • Problem: Optional variation process of a transformed martingale
  • Hints: Optional variation process of a transformed martingale

Hints: Optional variation process of a transformed martingale

Eksempel

Hints

You may use a similar approach as the proof of \((2.14)\) in the book. In particular use \((2.13)\) and \((2.8)\).

Relevante lenker

  • Eksempler:
    • Solution: Optional variation process of a transformed martingale