Introducing the Product Integral

(Feb 26, 2018)

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

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

Stephen covered the derivation of this in a previous blog, but I want to look more closely at the right-hand side of equation (1).  In particular, we can find an entirely different representation of \({}_tp_x\) as a product integral, which leads to many insights in survival models.

Recall how the integral in equation (1) is constructed.  Choose a partition of the interval \([0,t]\), that is some sequence \(\Delta_1,\Delta_2,\ldots,\Delta_n\) of non-overlapping sub-intervals that exactly cover the interval.  Define…

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Tags: survival models, survival probability, force of mortality, product integral

From small steps to big results

(Feb 1, 2018)

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, \(\mu_x\):

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

Where does this relationship come from?  We start by extending the survival time by an amount, \(h\), and look at the \((t+h)\)-year survival probability:

\[{}_{t+h}p_x = {}_tp_x.{}_hp_{x+t}\qquad(2)\]

which is simply to say that in order to survive \((t+h)\) years, you first need to survive \(t\) years and then you need to survive a further \(h\) years.  Of course, surviving \(h\) years is the same as not dying in \(h\) years, so equation (2) can be written thus:


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Tags: survival probability, force of mortality, differential equation

Why use survival models?

(Jan 4, 2012)

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.

We are sometimes asked why we prefer using μx, to which the lazy answer would be that this is what the CMI Technical Standards Working Party recommends, and it is how the the CMI has graduated all its tables since the early 1990s. Using μx to model mortality has a number of advantages, but here we will illustrate the simplest one.

One immediate advantage…

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Tags: survival analysis, survival models, force of mortality, hazard rate

A/E in A&E

(Apr 25, 2010)

We have often written about how modelling the force of mortality, μx, is superior to using the rate of mortality, qx.  This is all very well when you are building a formal model, but what about when you just want to quickly compare rates?  As it happens, the μx approach is quicker and more reliable, especially for portfolios with competing risks.

Consider a portfolio of term assurances where the policyholder can either lapse the policy or die.  For simplicity we will assume that each policyholder has only one policy, although in practice this is not the case and deduplication is required.  Suppose you want to compare the mortality rates between two portfolios which have very different lapse rates. …

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Tags: competing risks, force of mortality

Lost in translation

(Dec 30, 2009)

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 Ecx The time exposed to risk of dying at age x.
curve of deaths tpxmuxplust Probability density function for the future lifetime of an individual currently alive and aged exactly x.
force of mortality mux


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Tags: central exposed-to-risk, curve of deaths, force of mortality, initial exposed-to-risk, mortality law, mortality rate, survival rate, waiting time, survival models

Out for the count

(Jul 31, 2009)

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.  This violates the independence assumption and leads to parameter bias (amongst other undesirable consequences). If someone has three records aged 60, 61 and 62 initially, then these are not independent trials: the mere existence of the record at age 62 tells you that there was no death at age 60 or 61.

Life-company data often comes as a series of in-force extracts, together with a list of movements.  The usual procedure is to re-assemble the data to create a single record for each policy, using the policy number…

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Tags: survival models, force of mortality, GLM, missing data

Accelerating improvements in mortality

(Mar 19, 2009)

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 process.

We take the mortality hazard rates for males in England and Wales and calculate the relative mortality improvement, i.e. the improvement at age x in year y is:

1 - μx,y / μx,y-1

where the rates μx,y have been locally…

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Tags: mortality improvements, force of mortality

Competing risks

(Oct 20, 2008)

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.  Of course, if you have huge volumes of data, you may not lose too much by modelling qx.

A crucial caveat here is that the risk you are modelling is the only means of exit from the population, i.e. what is known as a single-decrement model.  If you have competing risks, however, then modelling qx becomes a lot more complicated.  In this situation, having large volumes of data will not help you at all.  In contrast, survival models can be fitted to competing risks without any additional assumptions.

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Tags: force of mortality

Survival models v. GLMs?

(Aug 12, 2008)

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.

In fact, survival models and GLMs are not necessarily mutually exclusive. It is true that GLMs are more commonly used for modelling the rate of mortality, qx, whereas survival models are always used for modelling the force of mortality, μx. Indeed, a survival model can be defined as a model for μx.

However, there are GLMs for the force of mortality as well. One notable example is the Poisson model for the number of deaths,…

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Tags: survival models, GLM, force of mortality

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