## F73ZF2 Survival Models II

Lecturer: S. J. Richards

#### Aims

• 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.

#### Summary

• 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.

#### Learning outcomes

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.

#### Assessment

The modules F73ZF2 and F73ZH3 are examined synoptically in a single three hour written examination paper.

#### Help

If you have any problems or questions regarding the module, you are encouraged to contact the lecturer by email at ????@ma.hw.ac.uk.

#### Module web page

Further information and course materials are available at http://www.ma.hw.ac.uk/~iain/teaching/mortality/mortality.html

#### Detailed syllabus

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