Angus Macdonald
Articles written by Angus Macdonald
The Curse of Cause of Death Models
Stephen's earlier blog explained the origin of the very useful result relating the life-table survival probability \({}_tp_x\) and the hazard rate \(\mu_{x+t}\), namely:
\[ {}_tp_x = \exp \left( - \int_0^t \mu_{x+s} \, ds \right). \qquad (1) \]
To complete the picture, we add the assumption that the future lifetime of a person now aged \(x\) is a random variable, denoted by \(T_x\), and the connection with expression (1) which is:
Introducing the Product Integral
Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:
\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]
Everything points to Poisson
One recurring theme in our forthcoming book, Modelling Mortality with Actuarial Applications, is the all-pervading role of likelihoods that suggest the lurking presence of a Poisson distribution. A popular assumption in modelling hazard rates is that the number of deaths observed at any given age is a Poisson random variable, so perhaps that might explain it?