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Counting Processes

Introduction of the course. Statistical analysis and comparisons of events evolved with time are central to biomedical research, and they are subjects of intensive investigations in biostatistics. Their underlying stochastic models involve counting processes $ N(t)$ of events and $ Y(t)$ of cases at risk, their hazard functions $ \lambda(t)$, and ultimately the construction of martingales $ dM(t) = dN(t) - Y(t)\lambda(t)dt$. A remarkably successful idea of martingale transform $ d\hat{B}(t) = H(t)dM(t)$ unifies various statistics $ \hat{B}(t)$ developed for many different statistical methods in survival analysis. In this short course we discuss foundations and properties of general counting processes and martingale framework. Then we survey survival analysis in the context of biostatistics, and examine the application of martingale framework for longitudinal data, including Kaplan-Meier estimate for survival functions and linear rank statistics for comparison of longitudinal data.

Tentative schedule. I taught this subject as a part of graduate course, aiming mostly at engineering graduate students who are not necessarily knowledgeable in the area of probability theory and statistics. Lecture is problem-oriented, and our goal is to complete all the problems presented in the class.

  1. Stochastic models and their distributions. Note #1 (last revised on June 22, 2016).
  2. Poisson processes and their properties. Note #2.
  3. Filtration and martingale in stochastic processes. Note #3.
  4. Censorship model for longitudinal data. Note #4.
  5. Comparison of two groups in survival analysis: Linear rank statistics. Note #5.

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