Angus Macdonald

I must have been one of many students who chose maths over medicine, because I have a terrible memory, and medics have to memorize books by the kilogram. In maths, if you understand how to do something, there is nothing to remember.  Right?

Up to a point.  Here are three mathematical relations where the order or direction matters, that I can never remember, no matter how often I have encountered them.

You are in charge of systems programming for an insurer writing disability insurance.  It is your job to write reporting modules to meet the needs of the actuaries, claims managers, accountants and so on.  Where to start?

The data would seem to be a good place.  I'll take it as read what kind of data the business will generate.  The question is how to represent it for efficient use in our programs - something we worry about so that the user doesn't have to.

In the course of a recent investigation, with my colleagues Dr Oytun Haçarız and Professor Torsten Kleinow, a key parameter was the mortality rate of persons suffering from Hypertrophic Cardiomyopathy (HCM), an inherited heart disorder characterized by thickening of the left ventricular muscle wall.  It is quite rare, so precision is not to be expected, and indeed an annual mortality rate of 1% \((q_x=0.01)\), independent of age \(x\), is widely cited.  I

In his recent blog Stephen described some empirical evidence in support of his practice of discarding the most recent six months' data, to reduce the effect of delays in reporting deaths. This blog demonstrates that the practice can also be justified theoretically in the survival modelling framework, although the choice of six months as the cut-off remains an empirical matter.

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:

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:

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

Most of the survival models used by actuaries are smooth or piecewise smooth; think of a Gompertz model for the hazard rate, or constant hazard rates at individual ages. When we need a cumulative quantity, we use an integral, as in the cumulative hazard function, \(\Lambda_x(t)\):

\[ \Lambda_x(t) = \int_0^t \mu_{x+s} \, ds. \qquad (1) \]