Survival Models (MATH3085/6143)

Chapter 10: Modelling Human Lifetime

André Victor Ribeiro Amaral
a.v.ribeiro-amaral@soton.ac.uk

University of Southampton

13/11/2025

Recap

In the last chapter, we

  • Derived the likelihood for an individual transition history and showed that the MLE for the time-homogeneous intensity is simply the total number of transitions from \(k\) to \(\ell\) divided by the total observed holding time in \(k\), i.e., \[ \hat{\mu}_{k\ell} = \frac{n_{k\ell}}{t^+_k}, \qquad k \neq \ell. \]

  • Derived the standard errors for \(\hat{\mu}_{k\ell}\) and constructed confidence intervals around our estimates.

  • Generalised the likelihood for multiple histories of transitions corresponding to individuals \(i = 1, \cdots, n\).

Chapter 10: Modelling Human Lifetime

Introduction

For the rest of this module our focus will be on modeling human life length, that is the time (from birth or some other specified time origin) to death in a specified human population.

Models for human human lifetimes have a huge importance, for example

  • Billions of pounds are invested in pension funds. Calculation of the liabilities requires us to be able to predict lifetimes of current and future pensioners.
  • Planning public services requires requires us to predict age-structured populations, sometimes within a small geographical area. This requires us to be able to forecast mortality (along with fertility and migration).

Special features of human lifetime

There are a number of practical reasons why human lifetime modelling requires special treatment, e.g.,

  • The long time scales involved (much longer than the horizon of a standard statistical study).
  • The requirement to use secondary data (collected for other purposes, such as census and death registration data)
  • Data sets are often large (good) but the data can be coarse (bad). For example, ages may only be provided in whole years.
  • Standard distributions tend not to provide a good fit for human lifetimes.
  • The distribution of human lifetime is changing (lifetimes are getting longer).

Models

As previously, we use \(T\) to denote the time from a specified origin (usually birth) until death.

Then we know that a survival model can be specified by the survival function \(S_T(t)\) or the hazard function \(h_T(t)\) for \(T\).

Alternatively, we can think of human survival as a two-state process for \(Y_x\in\{1,2\}\) (alive and dead) with an absorbing state.

Then the process is specified by the transition intensity function \(\mu_{12}(x)\) at time \(x\) or, alternatively, the transition probabilities \(p_{11}(x,t)=1-p_{12}(x,t)\).

Models are equivalent (Gap on page 114)

Recall that

\[ \begin{eqnarray*} p_{11}(x,t)\;=\; P(Y_{x+t}=1\,|\,Y_x=1)&=&P(T> x+t\,|\,T>x)\\[2pt] &=&S_T(t|x)\\[2pt] &=&\frac{S_T(x+t)}{S_T(x)}, \end{eqnarray*} \] where \(S_T(t|x)\) is the residual survival function (see Section 3.3.2). So, \(p_{11}(0,t)=S_T(t)\)

Similarly, \[ \mu_{12}(x)=\lim_{\delta t\to 0} \frac{p_{12}(x,\delta t)}{\delta t} =\lim_{\delta t\to 0} \frac{P(T\le x+\delta t\,|\,T>x)}{\delta t } =h_T(x). \]

So the models are equivalently specified.

Alternative notation (Gap on page 115)

Because of the importance of human lifetime modelling in demography and actuarial science, a specific notation has been developed.

\[ p_{11}(x,t)=S_T(t|x)\;\equiv\;\,_tp_x \]

and \[ p_{12}(x,t)=1-S_T(t|x)=1-\,_tp_x\;\equiv\;\,_tq_x. \]

So,

  • \({}_tp_x\) is probability of probability of survival to \(x+t\) given survival to age \(x\).
  • \({}_tq_x\) is the probability of death in \((x,x+t]\) given survival to age \(x\).

\(p_x\) and \(q_x\) notation (Gap on page 116)

When \(t=1\) (e.g., another year), the first subscript is usually omitted, so

  • \(q_x\) is the probability of death in \((x,x+1]\) given survival to age \(x\)
  • \(p_x= 1 - q_x\) is probability of survival to \(x+1\) given survival to age \(x\).

This is most commonly used when \(x\) is an exact integer number of years.

Note that if \(t\) is an integer then \[ \begin{eqnarray*} {}_tp_x &=\frac{S_T(x+t)}{S_T(x)} &= \frac{S_T(x+1)}{S_T(x)} \times \frac{S_T(x+2)}{S_T(x+1)} \times \cdots \times \frac{S_T(x+t)}{S_T(x+t-1)}\\[10pt] &&=p_{x}\times p_{x+1}\times \cdots \times p_{x+t-1} \end{eqnarray*} \]

and so \[ {}_tq_x=1- \left[(1-q_{x})\times (1-q_{x+1}) \times \cdots \times (1-q_{x+t-1})\right]. \]

Force of mortality (Gap on page 117)

We also have that \[ \mu_{12}(x)=h_T(x)\;\equiv\;\mu_x, \] which is called (in human lifetime applications) the force of mortality at age \(x\).

The force of mortality \(\mu_x\) is the limiting death rate at age \(x\), and as such is only constrained by \(\mu_x\ge 0\).

In practice, if we are measuring time in years, \(\mu_x<1\) for all except very extreme ages \(x\).

Lastly, the functions \({}_tp_x\), \({}_tq_x\) and \(\mu_x\) are related by \[ {}_tp_x\;=\;1-{}_tq_x\;=\;\exp\left(-\int_x^{x+t} \mu_x \mathrm{d}x\right). \]