A note on Whittaker–Henderson graduation: Bisymmetry of the smoother matrix

Abstract

Whittaker–Henderson (WH) graduation is a popular smoothing method that has been used for mortality table construction in the actuarial sciences and for the trend-cycle decomposition in time series econometrics. This paper proves that the smoother matrix of WH graduation is bisymmetric (i.e., symmetric centrosymmetric). This result implies, for example, that the first row of the smoother matrix is equivalent to the last row of it in reverse order. We also provide some related results.

Keywords:

• Whittaker–Henderson graduation
• smoother matrix
• bisymmetry
• centrosymmetry
• Hodrick–Prescott filter

2010 Mathematics Subject Classification:

• 62G05

JEL Classification:

• C221

Acknowledgments

The author thanks an anonymous referee for his/her valuable comments. The usual caveat applies.

Notes

1 These matrices are equivalent for p = 1, 2.

2 Although the corollaries of Proposition 3.1 corresponding to Corollaries 2.3 and 2.4 are also obtainable, they are omitted to save space.

Additional information

Funding

The Japan Society for the Promotion of Science supported this work through KAKENHI Grant Number 16H03606.