A state space formulation of Whittaker graduation, with extensions

• Published in: Insurance, mathematics & economics mathematics & economics, 1993-09, Vol.13 (1), p.7-14
• Main Author: Verrall, R.J.
• Format: Article
• Language: English
• Subjects: Generalised linear models; Graduation; Kalman filter; Life insurance; Linear models; Mathematical analysis; Mathematical models; Methods; Regression analysis; State space models
• ISSN: 0167-6687; 1873-5959
• DOI: 10.1016/0167-6687(93)90529-X

The methods used to graduate life tables can be divided into two broad categories: parametric curve fitting and non-parametric smoothing. Whittaker graduation is usually regarded as an example of a non-parametric regression analysis and is thus be placed in the second category. The purpose of this paper is to show that Whittaker graduation is equivalent to a dynamic regression analysis, in which one parameter of the fitted line is allowed to vary stochastically. It is therefore actually a combination of a parametric method and a smoothing method. The graduation can be performed using the Kalman filter, once it has been written in state space form. The state space form gives greater insights into the structure of Whittaker graduation, and into the assumptions made about the graduated values. It also suggests an automatic method of estimating the smoothing parameter, which is, at present, chosen subjectively by the graduator. Lastly, and perhaps most importantly, it shows that there is a close connection with parametric curve fitting methods, and implies that the form of Whittaker graduation can be improved.