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Osculatory Interpolation by Central Differences; with an application to Life Table Construction

Published online by Cambridge University Press:  18 August 2016

James Buchanan
James Buchanan
Affiliation:
Institute of Actuaries Faculty of Actuaries
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Extract

The new method described in Mr. King's recent paper, “On the Construction of Mortality Tables from Census Returns and Records of Deaths”, marks such a great advance on that employed in the construction of the official English Life Table, that it occurred to me that it might be worth examining whether the numerical application could not be further simplified by the use of central differences. Everett's formula (J.I.A., xxxv, 452) lends itself admirably to the construction of tables by subdivision of intervals. It involves only even central differences of each of the two middle terms of the series between which the interpolation has to be made, and, as was pointed out by the author in communicating his formula to the Journal, “each sum of three terms does double duty, serving both for “the preceding and the succeeding interval. In an extended “computation, the number of ‘sums of three terms’ to be “calculated is accordingly practically identical with the number “of intervals, and the labour of calculation is only about half “what it appears to be on the face of the formula.”

The problem before us is that of fitting between consecutive pairs of a series of points a series of partial interpolation curves, which shall have the same slope and curvature at their points of junction. In what follows, the central difference notation is that introduced by Dr. W. F. Sheppard in his paper on “Central Difference Formulæ” (Proceedings of the London Mathematical Society, xxxi, 449).

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 42 , Issue 3 , July 1908 , pp. 369 - 400
Copyright
Copyright © Institute and Faculty of Actuaries 1908

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References

page 370 note * In a paper on “A Practical Interpolation Formula” (Trans. Act. Soc. of America, ix, 211), Mr. Robert Henderson employs the symbols ρ and σ in the same senses as δ and μ. The latter has to actuaries a well-defined meaning, and on that ground it might be desirable to write σ or ½ σ for μ; but σ has been appropriated in Sheppard's notation (loc. cit. p. 474) to represent the operation inverse to that indicated by δ(σ = δ -1 ), and it is thought that the use of μ in the central difference scheme cannot as a rule lead to confusion.

page 373 note * Another mode of applying the conditions of osculation was indicated by Mr. Lidstone in the discussion following the reading of Mr. King's paper. The method described here was worked out before his remarks appeared in print, and as it has the advantage of showing how the coefficients of the final term or terms are built up of the coefficients of the unadjusted formula, and so of showing the error introduced by osculation, the work has been allowed to stand as originally executed. Mr. Lidstone, to whom I am indebted for some valuable criticisms, has sent me his alternative demonstration, which I have appended to this paper in the form of an additional note (p. 394).

page 377 note * It has been suggested that it would be useful if the coefficients were set out for decennial interpolations, which are so frequent in census works. A reference to Everett's paper (J.I.A., ixxv, 454) will show that the coefficients of the Everett formula for the odd interpolated values are inconvenient for numerical calculation, and those of the osculatory formula are equally so. The merit of the quinquennial interpolation, that the coefficients of the second and fourth differences are easy multiples of 8, would therefore be lost. By following the method of the paper and bisecting the decennial interval by Bessel's formula (19), which gives the same value for the middle of the interval as the osculatory formula, we can complete the interpolations for each half by means of the easy coefficients. Besides, in an extended Everett interpolation, there is always one set of “sums of three terms” which does duty only once, so that there is a further advantage in working with the shorter interval. The effect of the osculatory method is to replace the discontinuity error of the ordinary interpolation by a series of ripples, whose period is equal to the interval, and the height of the crests of these ripples will almost certainly be smaller with the quinquennial than with the decennial interpolation.

page 394 note * Vide Mr. Lidstone's remarks in the discussion on Mr. King's paper, pp. 283-5, supra.