# Angus Macdonald

### Articles written by Angus Macdonald

### Anglo-Saxon attitudes

*Scene: A meeting room, London, c.1997. Two actuaries are contemplating a flipchart on which is displayed some mathematics, including a double integral.*

*Actuary 1: "That's the kind of thing a Danish actuary would understand.'"*

*Actuary 2: "Yes, but could they calculate a premium rate?'"*

### No calculation without representation

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.

### The Mystery of the Non-fatal Deaths

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. It has foun

### A Model for Reporting Delays

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.

### From magical thinking to statistical thinking

### See You Later, Indicator

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:

### Right-Censoring Rules!

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:

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

### The long shadow of the life table

### Smooth Models Meet Lumpy Data

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*, \(\Lambda_x(t)\):

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

### More than one kind of information

This collection of blogs is called *Information Matrix*, and it is named after an important quantity in statistics. If we are fitting a parametric model of the hazard rate, with log-likelihood:

\[ \ell( \alpha_1, \ldots, \alpha_n ) \]

as a function of \(n\) parameters \(\alpha_1, \ldots, \alpha_n\), then the information matrix is the matrix of second-order partial derivatives of \(\ell\). That is, the matrix \({\cal I}\) with \(ij\)th component: