Statistical Analysis of Counting Processes

1. One-Dimensional Counting Processes.- 1.1. Probabilities on (0,?].- 1.2. The definition of one-dimensional counting processes.- 1.3. Construction of canonical counting processes.- 1.4. Intensities for canonical counting processes.- 1.5. Martingale decompositions for canonical counting processes.- 1.6. Statistical models and likelihood ratios.- Notes.- Exercises.- 2. Multivariate Counting Processes.- 2.1. Definition and construction of multivariate counting processes.- 2.2. Intensities and martingale representations.- 2.3. Products of canonical counting processes.- 2.4. Likelihood ratios.- 2.5. Discrete counting processes.- Exercises.- 3. Stochastic Integrals.- 3.1. Processes and martingales on WE.- 3.2. Definition and basic properties of stochastic integrals.- Notes.- Exercises.- 4. The Multiplicative Intensity Model.- 4.1. Definition of the full Aalen model.- 4.2. Product models and sufficient reductions.- 4.3. Estimation in the Aalen Model.- 4.4. Estimation in Markov chains.- 4.5. The Cox regression model.- 4.6. Maximum-likelihood estimation in Aalen models.- Notes.- Exercises.- 5. Asymptotic Theory.- 5.1. A limit theorem for martingales.- 5.2. Asymptotic distributions of Aalen estimators.- 5.3. Asymptotic distributions of product-limit estimators.- 5.4. Comparison of two intensities.- Notes.- Exercises.- 1. The principle of repeated conditioning.- 2. Weak convergence.- References.