**Module Description**

Calculations in clinical trials, pensions and life and health insurance require reliable estimates of transition intensities/survival rates. This module covers the estimation of these intensities. The syllabus includes the following: Concepts underlying actuarial modelling; distribution and density functions of the random future lifetime, the survival function and the force of hazard. It also covers estimation procedures for lifetime distributions including censoring, Kaplan-Meier estimate, Nelson-Aalen estimate and Cox model; statistical models of transfers between multiple states; maximum likelihood estimators for the transition intensities; Binomial and Poisson models of mortality; estimation of age-dependent transition intensities; the graduation process; testing of graduations and measuring the exposed-to-risk.

On completion of the course students should be able to (learning outcomes):

* Describe the principles of actuarial modelling

* Describe non-parametric estimation procedures for the lifetime distribution, including censoring, the Kaplan-Meier estimate, Nelson-Aalen estimate and Cox regression model (proportional hazards model);

* Derive maximum likelihood estimators (and hence estimates) for the transition intensities in models of transfers between states with piecewise constant transition intensities;

* Describe the Binomial and Poisson models of mortality, deriving maximum likelihood estimators for the probability/force of mortality and compare with the Markov models;

* Describe how to estimate transition intensities depending on age, exactly or using the census approximation, including calculation of exposed to risk and specification of census formulae based on various age definitions;

* Describe and carry out tests for the consistency of crude estimates with a standard table or a set of graduated estimates;

* Describe the process of graduation and the advantages and disadvantages of the various methods.

**Syllabus**

Concepts of actuarial modelling

Describe why and how models are used, their benefits and limitations.

Explain the concept of survival models, describe the model of lifetime or failure time from age x as a random variable, state the consistency condition between the random variable representing lifetimes from different ages, define the distribution and density functions of the random future lifetime, the survival function, the force of mortality or hazard rate, and derive relationships between them. State the Gompertz and Makeham laws of mortality. Compute life tables and define the expected value and variance of the complete and curtate future lifetime and derive expressions for them. Define the curtate future lifetime from age x and state its probability function.

Binomial and Poisson models of mortality

Describe an observational plan in respect of a finite number of individuals observed

during a finite period of time, and define the resulting statistics, including the waiting

times, derive the likelihood function for constant transition intensities in a Markov model of transfers between states described. Derive maximum likelihood estimators for the transition intensities and state their asymptotic joint distribution.

Describe the Poisson approximation to the estimator.

Describe the Binomial model of the mortality of a group of identical individuals subject to no other decrements between two given ages.

Derive the maximum likelihood estimator for the rate of mortality in the Binomial

model and its mean and variance and describe the advantages and disadvantages of the multiple state model and the Binomial model, including consistency, efficiency, simplicity of the estimators and their distributions, application to practical observational plans and generality.

Exact or approximate estimations of transition intensities.

Describe the principle of correspondence and explain its fundamental importance in the estimation procedure. Specify the data needed for the exact calculation of a central exposed to risk (waiting time) depending on age and sex and calculate a central exposed to risk. Explain how to obtain estimates of transition probabilities, including in the single decrement model the actuarial estimate based on the simple adjustment to the central exposed to risk. Explain the assumptions underlying the census approximation of waiting times. Explain the concept of rate interval.

Describe how to test crude estimates for consistency with a standard table or a set of graduated estimates, and describe the process of graduation by parametric formula, standard table or graphical and state the advantages and disadvantages of each.

Explain the concept of survival models.

Recognize the characteristics of survival data, e.g. censoring and truncation.

Describe the various ways in which lifetime data might be censored.

Describe the Kaplan-Meier (or product limit) estimate of the survival function and the Nelson-Aalen estimate of the cumulative hazard rate compute it from typical data and estimate its variance. Determine the proper method to be used in analyzing time-to-event data (e.g., parametric, semi-parametric or non-parametric method).

Describe the Cox proportional hazard model, derive the partial likelihood estimate, and state its asymptotic distribution. Perform survival analysis using a computer statistical software package. Interpret computer outputs.

- Module Supervisor: Yanchun Bao
- Module Supervisor: Hongsheng Dai