**Lecturer:** S. J. Richards

- To understand the use of mathematical models of mortality, illness and other life history events in the study of processes of actuarial interest.
- To be able to estimate the parameters in these models, mainly by maximum likelihood.
- To apply methods of smoothing observed rates of mortality and to test the goodness-of-fit of the models.

- Estimation for lifetime distributions: Kaplan-Meier estimate of the survival function, estimation for the Cox model for proportional hazards.
- Statistical models for transfers between multiple states (e.g., alive, ill, dead), the multi-state Markov model, relationship between probabilities of transfer and transition intensities, estimation for the parameters in these models. The binomial and Poisson models of mortality.
- Tests of consistency of crude estimates of rates of mortality and their graduated values.

At the end of studying this module, students should be able to:

- Estimate a survival function using the Kaplan-Meier method.
- Find the partial likelihood function in the Cox model.
- Use the partial likelihood to estimate the parameters (with standard errors) in the Cox model.
- Write down an appropriate Markov multi-state model for a system with multiple transfers.
- Obtain the Kolmogorov forward equations in a Markov multi-state model.
- Derive the likelihood function in a Markov multi-state model with data.
- Use the likelihood function to estimate the parameters (with standard errors) in a Markov multi-state model with data.
- Obtain the likelihood function in the 2-state model with states
*Alive*and*Dead*under the binomial or Poisson models. - Use any of three assumptions (uniform distribution of deaths, constant force of mortality, Balducci assumption) to reduce the binomial likelihood to a function of a single parameter, and estimate the parameter.
- To apply the test, the standardised deviations test, the sign test, the change of sign test, the grouping of signs test, the serial correlation test to testing the adherence of a graduation to data.

The course book is: I D Currie, *Survival Models*. The book is
essential reading and is available from the department. It contains
outline copies of the lecture material, all tutorial material and
copies of past examination papers.

Detailed syllabus

- Introduction, Notation and Revision:
- life time distributions, survival functions, rates and forces of mortality

- Estimating the Lifetime Distribution:
- cohort studies
- censoring
- Kaplan-Meier estimate of the survivor function
- Cox regression model, partial likelihood, estimation

- Markov Models: Theory:
- computation of
- multi-state Markov models
- Kolmogorov forward equations

- Markov models: Data and Estimation:
- 2-state model
- maximum likelihood estimate (MLE) of the force of mortality
- score function and the maximum likelihood theorem
- properties of the MLE of the force of mortality
- likelihood and estimation in the multi-state model

- Binomial and Poisson Models of Mortality
- binomial model
- three assumptions: uniform distribution of deaths, Balducci, constant force of mortality
- likelihood and estimation for the binomial model
- actuarial estimate of
- Poisson model

- Graduation and Statistical tests:
- graduation process
- testing adherence to data
- test, standardised deviations test, sign test, change of sign test, grouping of signs test, serial correlation test

- This module is linked and examined with F73ZH3, Survival Models III.