In this article, I’m going to look at choosing between mortality models using Hirotugu Akaike’s information criterion, the AIC.
I’m going to run through – at a very high level – the rationale behind the AIC and its construction because
A/E diagnostics are important but, if we have any mortality experience data, we should be using it to develop a model that takes account of that data, even if it’s nothing more than a simple how-much-heavier-or-lighter-is-the-mortality-of-this-population-than-average model. Otherwise, we’re not making full use of available information.
There are lots of possible approaches, including complex parametric formulas designed to capture all typically observed effects. But I promised concision and so in this article I’ll expound what I think is simultaneously one of the most powerful general approaches and one of the simplest. And the beauty of it is: we’ve already done most of the work.
I think it’s a shame that the ‘log’ in ‘log-likelihood’ is so often presented as a technical convenience or a device for avoiding numerical under/overflow. Yes, it is definitely both of these things, but it is much more fundamental.
Expected log-probability, i.e. entropy, lies at the heart of information theory. And the concept of entropy itself is pervasive, having extended beyond thermodynamics, its original home, into quantum physics and general relativity, as well as information theory.
So, without further ado, let’s define log-likelihood for mortality experience data.
Why ‘A over E’?
‘A over E’ literally refers to ‘actual’ deaths divided by ‘expected’ deaths as a measure of how experience data compares with a mortality.
In practice, ‘A over E’ is often interpreted as meaning the whole statistical caboodle, which is how I’ll use it here.
In the previous article we defined experience data, variables and mortality with respect to that data, and the \(\text{A}\) (actual) and \(\text{E}\) (expected) deaths operators.
In this article we’ll put \(\text{A}\) and \(\text{E}\) to work.
This is the first in a series of articles outlining mortality experience analysis fundamentals, by which I mean estimating the underlying mortality for individuals in a defined group (e.g. members of a pension plan) from mortality experience data.
This will be fairly technical, but I’ll aim
Stephen Richards’ and Angus Macdonald’s paper ‘On Contemporary Mortality Models for Actuarial Use’ (due to be discussed at the Institute of Actuaries on 24 October 2024) makes the case for the following in mortality experience analysis1:
Use individual data if at all possible.
Work in continuous time and use instantaneous mortality rates, i.e. μ rather than q.
Consider mortality experience data as comprising a series of Bernoulli trials over infinitesimally small time periods.
The paper could be read as a polemic against actuaries who can’t help but think in terms of q and whose first instinct is to group all time and age-dependent data on annual grids. Which is fine by me – I agree with the thrust of the paper and, in particular, the above three points.
So, having welcomed the paper, I do have a few observations…
Over the past 15 years or so, I have periodically searched in vain for a proportional sans serif font that is both high quality in itself but also suitable for programming, i.e. writing computer code. I finally decided to stop waiting and to make one myself.
The result is the open source Lisnoti (/lɪzˈnəʊtiː/) font, which can be downloaded here. It is also the font used to typeset this blog.