Information Matrix

In Richards & Macdonald (2024) we advocate that actuaries use the Kaplan-Meier estimate of the survival curve.  This is not just because it is an excellent visual communication tool, but also because it is a particularly useful data-quality check.

How should we describe a lifestyle change that doubles our likelihood of suffering a major traffic accident? Oddly,  evidence from Scotland suggests the answer is "worth making". Let me explain.

The short answer for mortality work is that your Poisson model is never truly Poisson.  The longer answer is that the true distribution has a similar likelihood, so you will get the same answer from treating it like Poisson.  Your model is pseudo-Poisson, but not actually Poisson.

In a recent blog, I looked at the most fundamental unit of observation in a mortality study, namely an individual life.  But is there such a thing as a fundamental unit of modelling mortality?  In Macdonald & Richards (2024) we argue that there is, namely an infinitesimal Bernoulli trial based on the mortality hazard.

Actuaries denote with \({}_tp_x\) the probability that a life alive aged exactly \(x\) years will survive a further \(t\) years or more.  The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, \(\mu_x\):

\[{}_tp_x = e^{-H_x(t)}\qquad(1)\]

where \(H_x(t)\) is the integrated hazard:

\[H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).\]

In Macdonald & Richards (2024), Angus and I continue our long-standing advocacy for using individual records for mortality analysis, rather than grouped counts of lives.  One argument in our paper is that the individual life is the most irreducible unit of observation in mortality analysis.  After all, any group can be disaggregated into individuals, but further subdivision would just be dismemberment.

In a (much) earlier blog, Angus introduced the product-integral representation of the survival function:

\[{}_tp_x = \prod_0^t(1-\mu_{x+s}ds),\qquad(1)\]

A common approach to teaching students about mortality is to view survival as a Bernoulli trial over one year. This view proposes that, if a life alive now is aged \(x\), whether the life dies in the coming year is a Bernoulli trial with the probability of death equal to \(q_x\).  With enough observations, one can estimate \(\hat q_x\), which is the basis of the life tables historically used by actuaries.

The late Iain Currie was a long-time advocate of smoothing certain parameters in mortality models.  In an earlier blog he showed how smoothing parameters in the Lee-Carter model could improve the quality of the forecast.  As Iain himself wrote, "this idea is not new" and traced its origins to Delwarde, Denuit & Eilers (2007).

Impossibility has often featured in humourous fiction.  From Lewis Carroll's White Queen, who "believed as many as six impossible things before breakfast", to Douglas Adams' Restaurant at the End of the Universe, there is entertainment value in absurdity.