Spline interpolation of demographic data revisited

Abstract

Spline functions have been suggested in demographic research for interpolating age-specific data as they have desirablesmoothness optimality properties. However, difficulties arise when boundary conditions need to be satisfied. Anadditional problem is that age-specific demographic data functions are necessarily non-negative, requiring the interpolatingspline to be monotonic non-decreasing. In this paper we describe a simple and effective alternative that circumvents theseproblems. We show that natural cubic splines can be used to interpolate age-specific demographic data and ensure thatrelevant boundary conditions on second derivatives are satisfied, thus preserving the desirable optimality property of theinterpolating function without the need to increase the degree of the spline function. The method involves incorporating oneor two additional strategically placed knots with values estimated from the data. We describe how the method works forselected fertility, population, and mortality data.

Keywords

• natural cubic spline
• monotonic
• fertility
• population
• mortality