Contingent means in multi-life models

Abstract

Multi-life models are useful in actuarial science for studying life contingency. Contingent probabilities are well-understood by most actuaries and are discussed extensively in the existing actuarial literature. However, the mean of a life in a multi-life model involving order of deaths is often found to be rather challenging to interpret by most actuaries who do not understand measure-theoretic probability. Standard textbooks on actuarial science or statistics do not elaborate on the correct interpretation of contingent means, leaving the actuaries at risk of making a blunder. This paper presents the correct interpretation both heuristically and rigorously using a non-measure-theoretic language, so that actuaries will be aware of some common misconceptions and avoid pitfalls in their work. The primary audience of this paper is practicing actuaries, actuarial students and actuarial educators. So we have given several actuarial applications. We hope that applied statisticians also will find this paper useful.

Keywords:

• conditional expectation
• order of deaths
• time-to-failure random variable

Acknowledgements

The second author thanks his students in the first year graduate mathematical statistics course for participating in a group problem solving session. Thanks are also due to the anonymous referee and the associate editor for their comments which led to a substantial improvement on a previous draft.