Lost in translation
Actuaries have a long-standing habit of using different terminology to statisticians. This page lists some common terms used by actuaries in mortality work and their "translation" for a non-actuarial audience. The terms and notation are those used by actuaries in the UK, but in every country I have visited the local actuaries have used similar notation.
Table 1. Common actuarial terms and their definition for statisticians.
| Actuarial term | Actuarial notation | Statistical description |
|---|---|---|
| central exposed to risk | \(E^c_x\) | The time exposed to risk of dying at age \(x\). |
| curve of deaths | \({}_tp_x\mu_{x+t}\) | Probability density function for the future lifetime of an individual currently alive and aged exactly \(x\). |
| force of mortality |
\(\mu_x\)
| In models with a single mortality decrement, this refers to the instantaneous hazard rate for mortality at exact age \(x\). In models with multiple states, this refers to the instantaneous transition intensity for mortality. |
| initial exposed to risk | \(E_x\) | The number of individuals alive aged exactly \(x\) at the start of an investigation period. |
| mortality law | varies | A functional form for the instantaneous hazard rate or probability of death, e.g. the Gompertz Law is \(\mu_x=\exp(\alpha+\beta x)\). |
| mortality rate | \(q_x\) | The probability of death between ages \(x\) and \(x+1\) in a Bernoulli model for the number of deaths, given that a life is alive at exact age \(x\) at outset. |
| survival rate | \({}_tp_x\) | The probability of survival from age \(x\) to age \(x+t\), given a life is alive at exact age \(x\), i.e. the survival curve or survivor function. |
It is interesting to note the different traditions within particular disciplines. For example, actuaries almost always use \(\mu_x\) to denote the force of mortality. Engineers, however, call the same thing the failure rate and typically denote it by \(\lambda\). Statisticians, meanwhile, will call the self-same concept the hazard function.
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Over-dispersion
Actuaries need to project mortality rates into the far future for calculating present values of pension and annuity liabilities. In an earlier post Stephen wrote about the advantages of stochastic projection methods. One method we might try is the two-dimensional P-spline method with the simple assumption that the number of deaths at age i in year j follows a Poisson distribution (Brouhns, et al, 2002). Figure 1 shows observed and fitted log mortalities for the cross-section of the
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A related translation table for non-statisticians can be found at Lost in Translation Reprise http://www.longevitas.co.uk/site/informationmatrix/lostintranslationrep…
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