Information Matrix

"The true nature of the Poisson distribution will become apparent only in connection with the theory of stochastic processes\(\ldots\)"

Feller (1950)

Stephen's earlier blog explained the origin of the very useful result relating the life-table survival probability \({}_tp_x\) and the hazard rate \(\mu_{x+t}\), namely:

\[ {}_tp_x = \exp \left( - \int_0^t \mu_{x+s} \, ds \right). \qquad (1) \]

To complete the picture, we add the assumption that the future lifetime of a person now aged \(x\) is a random variable, denoted by \(T_x\), and the connection with expression (1) which is:

In recent years we have published a number of papers on stochastic mortality models. A particular focus has been on the application of such models to longevity trend risk in a one-year, value-at-risk (VaR) framework for Solvency II. However, while a small group of models has been common to each paper, there have been changes in the calculation basis, most obviously where updated data have been used.

Our paper, "A stochastic implementation of the APCI model for mortality projections", was presented at the Institute and Faculty of Actuaries in October 2017. There was quite a discussion of the role of constraints in the fitting and forecasting of models of mortality. This got me wondering if constraints weren't in fact a red herring. This blog is a short introduction to the results of my investigation into the role, or indeed the non-role, of constraints in modelling and forecasting mortality.

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]

Ancient Greek philosophers had a paradox called "The Ship of Theseus"; if pieces of a ship are replaced over time as they wear out until every one of the original components is gone, is it still the same ship? At this point you could be forgiven for thinking (a) that this couldn't possibly be further removed from mortality modelling, and (b) that I had consumed something a lot more potent than tea at breakfast.

In survival-model work there is a fundamental relationship between the \(t\)-year survival probability from age \(x\), \({}_tp_x\), and the force of mortality, \(\mu_x\):

\[{}_tp_x = \exp\left(-\int_0^t\mu_{x+s}ds\right).\qquad(1)\]