Plotter for Hermite age model for mortality

Hermite interpolation for age-related mortality uses four basis functions to smoothly transition from (x0, Intercept) to (x1, Oldest). As is usual for mortality work, we operate on a logarithmic scale, so Intercept is log(mortality) at age x0, while Oldest is log(mortality) at age x1.

Age log(mortality) © www.longevitas.co.uk

At its most basic the Hermite mortality model requires only two of the Hermite functions, h00 and h01, to describe a smooth curve for log(mortality) between two ages. This gives rise to four mandatory parameters:

Mandatory parameters:
x0 Lower bound of age range at which log(mortality) has the value of Intercept.
x1 Upper bound of age range at which log(mortality) has the value of Oldest.
Intercept Value of log(mortality) at age x0 and below, i.e. where log(mortality) crosses the y-axis at age 0.
Oldest Value for log(mortality) at age x1 and above.

The Hermite spline h10 controls the initial direction of log(mortality) as it leaves the youngest age, x0 while h11 controls the shape of log(mortality) as it approaches the oldest age, x1. This gives rise to two optional parameters for log(mortality):

Optional parameters:
AgeGradientYoungest Initial gradient of log(mortality) at age x0.
AgeGradientOldest Gradient of log(mortality) as x1 is approached. This is usually best left close to zero.

We do not need to use every optional parameter in a Hermite model for log(mortality). The table below sets out the four useful combinations:

Model Parameters
Hermite I Intercept, Oldest
Hermite II Intercept, Oldest, AgeGradientYoungest
Hermite III Intercept, Oldest, AgeGradientOldest
Hermite IV Intercept, Oldest, AgeGradientYoungest, AgeGradientOldest
For a more detailed explanation with an example fitted model, see this blog. In addition to the basic age pattern for log(mortality), one can also add an age-related time trend, selection effects and seasonal variation. For full details of the model and its implementation, see Richards (2019).