Statistical Analysis of Counting Processes

A first version of these lecture notes was prepared for a course given in 1980 at the University of Copenhagen to a class of graduate students in mathematical statistics. A thorough revision has led to the result presented here. The main topic of the notes is the theory of multiplicative intens­ ity models for counting processes, first introduced by Odd Aalen in his Ph.D. thesis from Berkeley 1975, and in a subsequent fundamental paper in the Annals of Statistics 1978. In Copenhagen the interest in statistics on counting processes was sparked by a visit by Odd Aalen in 1976. At present the activities here are centered around Niels Keiding and his group at the Statistical Re­ search Unit. The Aalen theory is a fine example of how advanced probability theory may be used to develop a povlerful, and for applications very re­ levant, statistical technique. Aalen's work relies quite heavily on the 'theorie generale des processus' developed primarily by the French school of probability the­ ory. But the general theory aims at much more general and profound re­ sults, than what is required to deal with objects of such a relatively simple structure as counting processes on the line. Since also this process theory is virtually inaccessible to non-probabilists, it would appear useful to have an account of what Aalen has done, that includes exactly the amount of probability required to deal satisfactorily and rigorously with statistical models for counting processes.

Inhalt

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.