Information Matrix

In Macdonald et al (2018, Section 2.5) we describe the importance of deduplication, i.e. the identification of individuals behind multiple policies.  This is a critical step for a statistical model, as lives can be regarded as independent, whereas the mortality experience of two or more policies written on the same life clearly are not.

In Macdonald & Richards (2025), Stephen and I pointed out some benefits of models built up from instantaneous Bernoulli trials by product-integration (both of which have featured in previous blogs).

As discussed in earlier blogs, trailblazing actuaries Benjamin Gompertz and William Makeham used parametric models for the mortality hazard. However, the data they worked with were typically grouped into wide age ranges, which involves a loss of information if mortality rates are continually increasing.

When we first wrote our survival-modelling software in late 2005, we had to decide how to represent dates for the purpose of calculating exposure times.  We decided to adopt a real-valued approach, e.g. 14th March 1968 would be represented as 1968.177596 (the fractional part is \(\frac{31+29+14}{366}\), since 1968 is a leap year).

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.

We have discussed biological measures of age in this blog previously, so I was interested to find that last month, research examined the relationship between depressed mood and biological aging.

In 1860 William Makeham published a famous paper. In it he extended Gompertz's mortality law to include a constant term:

\[\mu_x=e^\epsilon+e^{\alpha+\beta x}\qquad(1),\]

It is well-known that Isaac Newton invented the calculus, and that his laws of motion and gravitation launched mathematical physics. Some brilliant British physicists carried on the tradition; think of Hamilton, Maxwell, Kelvin* et al. But where were the mathematicians? The answer is: across the English Channel.

Binary oppositions come easily to the human mind. Good and evil. Joy and sadness. Chalk and cheese. But, attracted as we are to neat categories, one question is whether these clean absolutes always exist in reality?

Today is the 200th anniversary of Benjamin Gompertz's reading of his famous paper before the Royal Society of London.  Generations of actuaries and demographers are familiar with his law of mortality:

\[\mu_x = e^{\alpha+\beta x}\qquad(1),\]