This book gives a thorough introduction to martingale and counting process methods in survival analysis, thereby filling a gap in the literature. It is primarily suitable for researchers and students, whose main interest is in survival analysis, but who want to learn more about the counting process approach, which Odd Aalen [Ann. Statist. 6, 534-545 (1978; Zbl 0383.62057)] popularized about ten years ago. The common martingale theory literature is often too inaccessible for non- specialists, and therefore the present book is welcome.
The authors start by giving some motivating examples concerning censored failure time data, and continue by giving the necessary martingale and counting process theory for reformulating the examples in a counting process framework. Properties of standard test statistics and estimators are considered, and censored data regression models introduced. The martingale central limit theorems are used to derive large sample results, including Kaplan-Meier and weighted log rank statistics. The considered models and methods are generally nonparametric or semiparametric, ranging from the Kaplan-Meier estimator of the cdf of a single sample, to the Cox regression model.
Five appendices finish the book: Some measure theory is reviewed, as well as weak convergence of stochastic processes and the martingale central limit theorem. Finally, some data sets and exercises are included.