Statistical Models Based on Counting Processes

Modern survival analysis and more general event history analysis may be effectively handled within the mathematical framework of counting processes. This book presents this theory, which has been the subject of intense research activity over the past 15 years. The exposition of the theory is integrated with careful presentation of many practical examples, drawn almost exclusively from the authors'own experience, with detailed numerical and graphical illustrations. Although Statistical Models Based on Counting Processes may be viewed as a research monograph for mathematical statisticians and biostatisticians, almost all the methods are given in concrete detail for use in practice by other mathematically oriented researchers studying event histories (demographers, econometricians, epidemiologists, actuarial mathematicians, reliability engineers and biologists). Much of the material has so far only been available in the journal literature (if at all), and so a wide variety of researchers will find this an invaluable survey of the subject.

Table of Contents:
I. Introduction.- I.1 General Introduction to the Book.- I.2 Brief Survey of the Development of the Subject.- I.3 Presentation of Practical Examples.- II. The Mathematical Background.- II.1 An Informal Introduction to the Basic Concepts.- II.2 Preliminaries: Processes, Filtrations, and Stopping Times.- II.3 Martingale Theory.- II.4 Counting Processes.- II.5 Limit Theory.- II.6 Product-Integration and Markov Processes.- II.7 Likelihoods and Partial Likelihoods for Counting Processes.- II.8 The Functional Delta-Method.- II.9 Bibliographic Remarks.- III. Model Specification and Censoring.- III.1 Examples of Counting Process models for Complete Life History Data. The Multiplicative Intensity Model.- III.2 Right-Censoring.- III. 3 Left-Truncation.- III.4 General Censorship, Filtering, and Truncation.- III.5 Partial Model Specification. Time-Dependent Covariates.- III.6 Bibliographic Remarks.- IV. Nonparametric Estimation.- IV. 1 The Nelson-Aalen estimator.- IV.2 Smoothing the Nelson-Aalen Estimator.- IV.3 The Kaplan-Meier Estimator.- IV.4 The Product-Limit Estimator for the Transition Matrix of a Nonhomogeneous Markov Process.- IV.5 Bibliographic Remarks.- V. Nonparametric Hypothesis Testing.- V.1 One-Sample Tests.- V.2 k-Sample Tests.- V.3 Other Linear Nonparametric Tests.- V.4 Using the Complete Test Statistic Process.- V.5 Bibliographic Remarks.- VI. Parametric Models.- VI.1 Maximum Likelihood Estimation.- VI.2 M-Estimators.- VI.3 Model Checking.- VI.4 Bibliographic Remarks.- VII. Regression Models.- VII.1 Introduction. Regression Model Formulation.- VII.2 Semiparametric Multiplicative Hazard Models.- VII.3 Goodness-of-Fit Methods for the Semiparametric Multiplicative Hazard Model.- VII.4 Nonparametric Additive Hazard Models.- VII.5 Other Non- and Semi-parametric Regression Models.- VIL6 Parametric Regression Models.- VII.7 Bibliographic Remarks.- VIII. Asymptotic Efficiency.- VIII.1 Contiguity and Local Asymptotic Normality.- VIII.2 Local Asymptotic Normality in Counting Process Models.- VIII.3 Infinite-dimensional Parameter Spaces: the General Theory.- VIII.4 Semiparametric Counting Process Models.- VIII.5 Bibliographic Remarks.- IX. Frailty Models.- IX.1 Introduction.- IX.2 Model Construction.- IX. 3 Likelihoods and Intensities.- IX.4 Parametric and Nonparametric Maximum Likelihood Estimation with the EM-Algorithm.- IX.5 Bibliographic Remarks.- X. Multivariate Time Scales.- X.1 Examples of Several Time Scales.- X.2 Sequential Analysis of Censored Survival Data with Staggered Entry.- X.3 Nonparametric Estimation of the Multivariate Survival Function.- X.4 Bibliographic Remarks.- Appendix The Melanoma Survival Data and Standard Mortality Tables for the Danish Population 1971-75.- References.- Author Index.