Right-Censoring Rules!

(Jan 23, 2019)

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:

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

In fact, all of classical life insurance mathematics follows from this assumption; see Dickson, Hardy and Waters (2013).  This is an example of a probabilistic model in action.  We specify a model in terms of one or more random variables and then calculate the probabilities of interesting events.

The inverse problem is the domain of statistics.  Given…

Read more

Tags: survival analysis, right-censoring, counting process

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…

Read more

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

Features of the survival curve

(Sep 10, 2008)

 The survival curve is simply the proportion of lives surviving to each age.  Below is an example for males at initial age 60 in the United Kingdom, using the Interim Life Table from the Government Actuary's Department:

Survival curve for males in United Kingdom between 2004 and 2006

The survival curve starts at 1 (or 100%) as everyone is alive at outset, and decreases monotonically towards zero (or 0%) as people die. The survival curve is better known to actuaries as tpx, the probability of a life aged x surviving to age x+t.  An oft-unappreciated feature of the survival curve is that the area underneath it is simply the life expectancy.

Instead of plotting the survival curve, exactly the same data can be used plot the distribution of age at death:

Distribution of age at death for males in United Kingdom between 2004 and 2006

The graph above is known to actuaries…

Read more

Tags: survival analysis, survival curve, curve of deaths

Are you allergic to statistical models?

(Aug 4, 2008)

Or do you know someone who is? Some people are uncomfortable with the idea of statistical models, especially ones with parameters. It is worth remembering that in 1958 Kaplan and Meier introduced the idea of an empirical survival curve, also called the product-limit estimator. The basic idea is to re-arrange the mortality experience data in such a way as to demonstrate the survival rates of different sub-groups. The key feature of the Kaplan-Meier curve is that there are no parameters involved: the empirical survival curve is simply a re-arrangement of the experience data, and involves no model fitting and no parameter estimation.

In the chart below we show the Kaplan-Meier curves for males and females in a…

Read more

Tags: survival analysis, Kaplan-Meier

Find by key-word

Find by date

Find by tag (show all )