Angus Macdonald
Articles written by Angus Macdonald
Stopping the clock on the Poisson process
"The true nature of the Poisson distribution will become apparent only in connection with the theory of stochastic processes\(\ldots\)"
Feller (1950)
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?