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Statistical models based on counting processes. (English) Zbl 0824.60003
Springer Series in Statistics. New York, NY: Springer-Verlag. xi, 767 p. (1995).
The book is devoted to mathematical theory of event history analysis (or of generalized survival analysis). Most of the results presented here were available only in the journal literature. The theory is built for observations recorded in continuous time. Very often the data are incomplete because of right-censoring, left-censoring, or both. The authors use modern mathematical tools (martingales, product-integration, contiguity, local asymptotic normality) for presentation and analysis of the models. Although precise definitions and statements (and sometimes also an informal explanation) are given, the book cannot be considered as a textbook. The results are illustrated on numerical examples coming almost exclusively from biostatistical experience of the authors. The examples concern survival with malignant melanoma, mortality of diabetics, suicides, software reliability, and other topics.
The chapters of the book are: I. Introduction; II. The mathematical background; III. Model specification and censoring; IV. Nonparametric estimation; V. Nonparametric hypothesis testing; VI. Parametric models; VII. Regression models; VIII. Asymptotic efficiency; IX. Frailty models; X. Multivariate time scales.
This impressive monograph will become a standard reference book in generalized survival analysis. The authors also indicate some recent lines of research. The book can be recommended to mathematically oriented specialists in this area, especially to biostatisticians or reliability engineers, and to postgraduate students.
Reviewer: J.Anděl (Praha)

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J25 Continuous-time Markov processes on general state spaces
60G07 General theory of stochastic processes
60G12 General second-order stochastic processes
62N05 Reliability and life testing