Iain Currie

The year 1825 was a significant one not only for actuaries but for the wider scientific community: Benjamin Gompertz published his landmark paper on the graduation of human mortality (Gompertz, 1825). There were at least three completely new ideas in his paper. First, he gave his famous law of mortality. To quote Gompertz:

This is the fourth and final blog on the use of constraints in the modelling and forecasting of mortality. The previous three blogs (here, here and here) demonstrated that there is no need to worry about which linear constraints to use: the fitted values of mortality and crucially their forecast values always come out the same.

I'm a statistician so I worry about standard errors just as much as I worry about point estimates. My blog Up close and intimate with the APCI model looked at the effect of different constraints on parameter estimates in models of mortality. This blog looks at the effect of constraints on the standard errors of the parameter estimates.

This blog brings together two pieces of work. The first is the paper we presented to the Institute and Faculty of Actuaries, "A stochastic implementation of the APCI model for mortality projections", which will appear in the British Actuarial Journal. The second is a previous blog where I examined the role of constraints in models of mortality.

Our paper, "A stochastic implementation of the APCI model for mortality projections", was presented at the Institute and Faculty of Actuaries in October 2017. There was quite a discussion of the role of constraints in the fitting and forecasting of models of mortality. This got me wondering if constraints weren't in fact a red herring. This blog is a short introduction to the results of my investigation into the role, or indeed the non-role, of constraints in modelling and forecasting mortality.

A population consists of individuals, each with their own genetics, lifestyle, and yes, their very own force of mortality. National mortality data, such as held by the Office for National Statistics (ONS), are observed only at the population level and the variation in the force of mortality across individuals of the same age is forever hidden. The purpose of this blog is to show how we can attempt to model this hidden heterogeneity.

Heligman and Pollard published a famous paper in 1980 with the title "The age pattern of mortality". In their paper they proposed an additive, three-component model of mortality:

\[q_x/p_x = f_I(x) + f_S(x) + f_A(x)\]