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.