Statistical Models Based on Counting Processes

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.