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On the Rationale of Formulæ for Graduation by Summation

Published online by Cambridge University Press:  18 August 2016

George J. Lidstone
George J. Lidstone
Affiliation:
The Equitable Life Assurance Society
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Extract

So much has been written on the algebra of formulæ for graduating an irregular series by means of successive summations, and on the theoretical errors introduced by this process, that there is possibly some danger of the underlying principles, on which all such formulæ are founded, being overlooked or insufficiently understood. In the present note an attempt is made to deal with this matter from first principles, with the object of rendering clear how such formulæ Work and why it is that they produce, with greater or less success according to the nature of the particular formula, a smooth succession of values from rough and irregular data.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 41 , Issue 3 , July 1907 , pp. 348 - 360
Copyright
Copyright © Institute and Faculty of Actuaries 1907

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References

page 349 note * For the present purpose a somewhat narrow view has to be taken of the proper function of graduation; for a much broader view, with which the present writer is in entire sympathy, see remarks by Hardy, G. F., J.I.A., xxxiii, 491.Google Scholar

page 354 note * In order to save space, only one-half of the curve is shown in each case, viz., the left-hand half (corresponding to the central and preceding terms) for Spencer's and G. F. Hardy's formulæ, and the right-hand half (corresponding to the central and following terms) for Woolhouse's and Higham's formulæ. In each case the curve is in fact continued symmetrically on the other side of the maximum ordinate.

page 356 note * In these diagrams, the straight line represents the average value of the unadjusted values. The graduated values tend in general to group themselves round the straight line; but in the particular case represented in Diagram B, the graduated values are seen to lie almost entirely above the straight line. This is because the average value of those ungraduated values for which graduated values are obtained, happens in this case to be appreciably greater than the general average of the whole ungraduated series, including the values at the beginning and end of the series, where no graduated values are obtainable.