# Information Matrix

## Filter Information matrix

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### Introducing the Product Integral

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

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

Written by: Angus Macdonald

### From small steps to big results

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, $$\mu_x$$:

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

Written by: Stephen Richards

### Why use survival models?

We and our clients much prefer to analyse mortality continuously, rather than in yearly intervals like actuaries used to do in previous centuries. Actuaries normally use μx to denote the continuous force of mortality at age x, and qx to denote the yearly rate of mortality. For any statisticians reading this, μx is the continuous-time hazard rate.
Written by: Stephen Richards

### A/E in A&E

We have often written about how modelling the force of mortality, μx, is superior to using the rate of mortality, qx.
Written by: Stephen Richards

### 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.

Written by: Stephen Richards

### Out for the count

In an earlier post we described a problem when fitting GLMs for qx over multiple years.  The key mistake is to divide up the period over which the individual was observed in a model for individual mortality.
Written by: Stephen Richards

### Accelerating improvements in mortality

In February 2009 a variation on the Lee-Carter model for smoothing and projecting mortality rates was presented to the Faculty of Actuaries.  A key question for any projection model is whether the process being modelled is stable.  If the process is not stable, then a model assuming it is stable will give misleading projections.  Equally, a model which makes projections by placing a greater emphasis on recent data will be better able to identify a change in tempo of the underlying p

Written by: Stephen Richards

### Competing risks

Survival models are models for continuous risk, e.g. the force of mortality, μx.  We showed in an earlier post why this is more powerful and efficient than modelling the rate of mortality, qx
Written by: Stephen Richards

### Survival models v. GLMs?

At some point you may be challenged to decide whether to use survival models or the older generalised linear models (GLMs). You could be forgiven for thinking that the two were mutually exclusive, especially since some commercial commentators have tried to frame the debate that way.

Written by: Stephen Richards