# 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

**Tags:**Filter information matrix by tag: survival models, Filter information matrix by tag: survival probability, Filter information matrix by tag: force of mortality, Filter information matrix by tag: product integral

### 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

**Tags:**Filter information matrix by tag: survival probability, Filter information matrix by tag: force of mortality, Filter information matrix by tag: differential equation

### Why use survival models?

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: survival analysis, Filter information matrix by tag: survival models, Filter information matrix by tag: force of mortality, Filter information matrix by tag: hazard rate

### A/E in A&E

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: competing risks, Filter information matrix by tag: force of mortality

### 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

**Tags:**Filter information matrix by tag: central exposed-to-risk, Filter information matrix by tag: curve of deaths, Filter information matrix by tag: force of mortality, Filter information matrix by tag: initial exposed-to-risk, Filter information matrix by tag: mortality law, Filter information matrix by tag: mortality rate, Filter information matrix by tag: survival rates, Filter information matrix by tag: waiting time, Filter information matrix by tag: survival models

### Out for the count

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: survival models, Filter information matrix by tag: force of mortality, Filter information matrix by tag: GLM, Filter information matrix by tag: missing data

### 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

**Tags:**Filter information matrix by tag: mortality improvements, Filter information matrix by tag: force of mortality

### Competing risks

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: force of mortality

### 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

**Tags:**Filter information matrix by tag: survival models, Filter information matrix by tag: GLM, Filter information matrix by tag: force of mortality