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Right-Censoring Rules!

A fundamental assumption underlying most modern presentations of mortality modelling (see our new book) is that the future lifetime of a person now age \(x\) can be represented as a non-negative random variable \(T_x\). The actuary's standard functions can then be defined in terms of the distribution of \(T_x\), for example:

\[{}_tp_x = \Pr[ T_x > t ].\]

Written by: Angus MacdonaldTags: Filter information matrix by tag: survival analysis, Filter information matrix by tag: right-censoring, Filter information matrix by tag: counting process

Mme Calment's other secret?

Favourite stories can, in the process of retelling, turn into legends. But might it eventually become difficult to distinguish between legend and myth? Indeed, are we longevity watchers about to lose a favourite story? Consider what follows, dear readers, and decide for yourselves...
Written by: Gavin RitchieTags: Filter information matrix by tag: longevity, Filter information matrix by tag: supercentenarians

The long shadow of the life table

For centuries, the life table has been at the centre of actuarial work.  It sets out the gradual extinction of a hypothetical population, often a birth cohort.
Written by: Angus MacdonaldTags: Filter information matrix by tag: life table

Hedging or betting?

Last week I presented at Longevity 14 in Amsterdam. A recurring topic at this conference series is index-based approaches to managing longevity risk. Indeed, this topic crops up so reliably, one could call it a hardy perennial.

Written by: Stephen RichardsTags: Filter information matrix by tag: basis risk, Filter information matrix by tag: concentration risk, Filter information matrix by tag: model risk

'D' is for deficiency

The United Kingdom has long had persistent regional disparities in mortality, and thus in life expectancy.
Written by: Stephen RichardsTags: Filter information matrix by tag: Scotland, Filter information matrix by tag: sunshine, Filter information matrix by tag: Vitamin D

Smooth Models Meet Lumpy Data

Most of the survival models used by actuaries are smooth or piecewise smooth; think of a Gompertz model for the hazard rate, or constant hazard rates at individual ages. When we need a cumulative quantity, we use an integral, as in the cumulative hazard function, \(\Lambda_x(t)\):

\[ \Lambda_x(t) = \int_0^t \mu_{x+s} \, ds. \qquad (1) \]

Written by: Angus MacdonaldTags: Filter information matrix by tag: Nelson-Aalen

Valuing liabilities with survival models

Regular readers of this blog will know that we are strong advocates of the benefits of modelling mortality in continuous time via survival models. What is less widely appreciated is that a great many financial liabilities can be valued with just two curves, each entirely determined by the force of mortality, \(\mu_{x+t}\), and a discount function, \(v^t\).

Written by: Stephen RichardsTags: Filter information matrix by tag: survival curve, Filter information matrix by tag: curve of deaths

More than one kind of information

This collection of blogs is called Information Matrix, and it is named after an important quantity in statistics. If we are fitting a parametric model of the hazard rate, with log-likelihood:

\[ \ell( \alpha_1, \ldots, \alpha_n ) \]

as a function of \(n\) parameters \(\alpha_1, \ldots, \alpha_n\), then the information matrix is the matrix of second-order partial derivatives of \(\ell\). That is, the matrix \({\cal I}\) with \(ij\)th component:

Written by: Angus MacdonaldTags: Filter information matrix by tag: information, Filter information matrix by tag: indicator process

Testing the tests

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.

Written by: Stephen RichardsTags: Filter information matrix by tag: deviance residuals, Filter information matrix by tag: autocorrelation, Filter information matrix by tag: Fisher transform