Stephen Richards

Examining residuals is a key aspect of testing a model's fit. In two previous blogs I first introduced two competing definitions of a residual for a grouped count, while later I showed how deviance residuals were superior to the older-style Pearson residuals. If a model is correct, then the deviance residuals by age should look like random N(0,1) variables.

Assume we have a random variable, \(X\), with expected value \(\eta\) and variance \(\sigma^2\). Often we find ourselves wanting to know the expected value and variance of a function of that random variable, \(f(X)\). Fortunately there are some workable approximations involving only \(\eta\), \(\sigma^2\) and the derivatives of \(f\). In both cases we make use of a Taylor-series expansion of \(f(X)\) around \(\eta\):

\[f(X)=\sum_{n=0}^\infty \frac{f^{(n)}(\eta)}{n!}(X-\eta)^n\]

[Regular visitors to our blog will have guessed from the title that this posting is about obesity. If you landed here looking for WWII material, you want the other Battle of the Bulge.]

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.

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, \(\mu_x\):

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

Back in the days before personal computers, actuaries relied solely on published tables for their calculations. These were not just the mortality tables, but monetary functions of these tables known as commutation factors. My old student tables from 1980 list commutation and other factors at discount rates of 4%, 6% and 8% (the latter rate seems almost comically high by current standards).