# How to model missing statistics in an actuarial table of age related mortality rates?

This is more of a historical related question. Edmond Halley in 1693 published an actuarial table showing mortality rates. The tables show yearly statistics for ages 1-84 and a summary statistic for ages 84-100. I have searched hard for the missing data, but with no luck (See also Geoffrey Heywood's article).

So this question has two parts:

• Can anyone point me to a different source showing the original paper?
• What would be the best model to generate the missing statistics between these years?
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As I see the table and the graph (I don't know if it's the right one, but suppose that, it is linked from ) the data for age 84+ are insufficient (the graph line starts to bend). They cleary know why they cut it there. I think I wouldn't try to extrapolate in this case. –  Tomas Oct 21 '12 at 10:54

Gompertz (1825) - log hazard linear in age - and Makeham (1860) models (adding a constant to that exponentially increasing hazard) were used pretty successfully for mortality in their time, at least at adult ages. They don't work as well now (but mortality has changed a lot).

My first inclination would be to try simple models like those for mortality, and check them for fit. The logit of $q_x$ should behave similarly to log-hazard. If you need the whole life table in a functional form, perhaps a Heligman-Pollard model or if just a good fit is needed something using splines (as is often used these days to model mortality).

There are whole books on modelling mortality; it's not a subject you can properly cover in ten lines.

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Maybe a sensible approach could be the Brass logit model. You calibrate $ln(p^{target}_t)=a+b*ln(p^{base}_t)$ being target and base an incomplete and a complete table. Then you can complete the missing $p_t^{target}$ thanks to the calibrated model.
You have not explained what you mean with $p^{target}_t$ and $p^{base}_t$. In the question the O.P. has only one mortality rate per year ($p_t$). –  Tomas Oct 21 '12 at 10:37