This paper surveys some statistical models of survival data. Competing risks models are described; the unidentifiability of net decrements suggests a sceptical approach to the use of underlying single decrement tables. Approaches based on observations of complete lifetimes (with censoring) are surveyed including the Kaplan-Meier and Nelson-Aalen estimates. Regression models for lifetimes depending on covariates are discussed, in particular the Cox model and partial likelihood estimation.

Counting processes and their compensators are introduced at a heuristic level. The martingale property of stochastic integrals with respect to a compensated counting process leads to moment estimates and asymptotic normal distributions for statistics arising in multiple state, non-parametric and semi-parametric models. The place of survival models in actuarial education is discussed.

A new class of reliability point-process models for dependent components is introduced. The dependence is expressed through a regression, following a form suggested by Cox (1972) for survival data analysis involving the current life-length of the components. After formulating the current-life process as a Markov process with stationary transitions and stating some general results on asymptotic behavior, we describe the stationary distributions in some bivariate examples. Finally, we discuss statistical inference for the new models, exhibiting and justifying full- and partial-likelihood methods for their analysis.

The estimation of the parameters of a discrete-time semi-Markov process is considered, when the data consist of records of a large number of individuals observed in a specified short period of time. Such a problem may arise in the modelling of intra-urban mobility. Methods of estimation of the parameters are suggested, and a numerical example discussed for the case when the observation period consists of three consecutive time points.