Information Matrix

If this and other recent blogs have a historical flavour, the reason is the 200th anniversary of the 1825 paper by Benjamin Gompertz that introduced his eponymous law of mortality. In the course of his own researches, Stephen drew to my attention a short letter published by Wilhelm Lazarus in Journal of the Institute of Actuaries in 1862. It is a remarkable document.

Stephen and I recently presented a pair of papers to the Institute and Faculty of Actuaries: Richards & Macdonald (2024) and Macdonald & Richards (2024).  In these papers we encourage actuaries to use continuous-time models in their work. But where does that leave discrete-time?

In a recent blog, I looked at the most fundamental unit of observation in a mortality study, namely an individual life. But is there such a thing as a fundamental unit of modelling mortality?  In Macdonald & Richards (2024) we argue that there is, namely an infinitesimal Bernoulli trial based on the mortality hazard.

In a (much) earlier blog, Angus introduced the product-integral representation of the survival function:

\[{}_tp_x = \prod_0^t(1-\mu_{x+s}ds),\qquad(1)\]

Our new book, Modelling Mortality with Actuarial Applications, describes several non-parametric estimators of two quantities:

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]