Information Matrix

Mortality at post-retirement ages has three apparent stages:

1. A broadly Gompertzian pattern up to age 90 (say), i.e. the mortality hazard is essentially linear on a logarithmic scale.

2. The rate of increase in mortality slows down, the so-called "late-life mortality deceleration".

A recurring feature in my previous blogs, such as this one on information, is the indicator process:

\[Y^*(x)=\begin{cases}1\quad\mbox{ if a person is alive at age \(x^-\)}\\0\quad\mbox{ otherwise}\end{cases}\]

where \(x^-\) means immediately before age \(x\) (never mind the asterisk for now). When something keeps cropping up in any branch of mathematics or statistics, there are usually good reasons, and this is no exception. Here are some:

This blog brings together two pieces of work. The first is the paper we presented to the Institute and Faculty of Actuaries, "A stochastic implementation of the APCI model for mortality projections", which will appear in the British Actuarial Journal. The second is a previous blog where I examined the role of constraints in models of mortality.

A fundamental assumption underlying most modern presentations of mortality modelling (see our new book) is that the future lifetime of a person now age \(x\) can be represented as a non-negative random variable \(T_x\). The actuary's standard functions can then be defined in terms of the distribution of \(T_x\), for example:

\[{}_tp_x = \Pr[ T_x > t ].\]

Happy New Year to all our readers!