Summation and Some Other Methods of Graduation – the Foundations of Theory

Abstract

SynopsisSeveral methods of graduation depend on linear operators–that is, operators of the type …+λ−nE−n+‥+λ0+…+λnEn+… Few attempts have been made to compare these methods, and there is no general theory. There is need for such a theory, for it is easy to show that unless some relation is assumed to hold between neighbouring graduated rates, there is no justification for graduating at all. Any such theory must explain this relation, which cannot be of the simple type that leads to curve fitting.The paper has two objects : to compare and judge these methods, and to devise a general theory applying to them all.The first approach is by the classical method–separate examination of fidelity and smoothness.