A Bayesian interpretation of Whittaker—Henderson graduation

Abstract

The primary purpose of the paper is to place Whittaker—Henderson graduation in a Bayesian context and show that this determines in a precise manner the extent to which goodness-of-fit should be traded off against smoothness in the Whittaker—Henderson loss function. This is done in Section 2. Section 3 generalizes the set of admissible graduating functions to a normed linear space. A specific example of this generalization is Schoenberg graduation, which is strongly related to Whittaker—Henderson but leads to particular spline graduations. These are placed in a Bayesian context parallel to that of Section 2. The similarity to shrunken or Stein-type estimators is pointed out. Section 4 considers the practical implications of these theoretical developments. Transformations of observations under graduation are examined and shown to be natural in some circumstances. The precise trade off mentioned above is enlarged upon, and the main conclusions reached here are seen to carry over to general spline graduation. The relation between Whittaker—Henderson and spline graduation is identified.