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On the Error introduced into Mortality Tables by Summation Formulas of Graduation
Published online by Cambridge University Press: 18 August 2016
Extract
The controversy which took place a few years ago (J.I.A. xxvi, 77 and 420 ; xxix, 59, 232 and 236) between Dr. Sprague and the late Mr. Woolhouse on Summation Formulas for the graduation of mortality tables, and more particularly on that of Mr. Woolhouse himself, left the question in a very unsatisfactory state, as a few extracts from the writings of these two great, but divergent, authorities will sufficiently show:
Dr. Sprague, xxvi, 111, said, “We must include in one “general condemnation all such graduation formulas as “Mr. Woolhouse's, Mr. Higham's, and Mr. Ansell's. They all “have a tendency to distort the true law of the facts”; and, again, xxix, 61, “I see no sufficient reason for distinguishing “between his (Mr. Woolhouse's) formula, and others of a like “kind ; and I include them all in one general condemnation.”
In contradiction of this, Mr. Woolhouse, xxvi, 424, said, “The method (his own) stands alone, as systematically based on “rational principles”; and, again, xxix, 241, “I emphatically” repeat that, with practicable data, the application of my system “of adjustment to a mortality table will always give the best ”possible results.”
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- Research Article
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- Copyright © Institute and Faculty of Actuaries 1907
References
page 59 note * See J.I.A., xxxii, 387, where Mr. Todhunter proves the converse proposition, that the average of the interpolated values given by any odd number of curves can be expressed by a summation formula.
page 64 note * It should be noted that the values of colog 10 Px, calculated by the constants employed in Mr. G. F. Hardy's graduation of the OM(5) Table, are given, to seven places of decimals, on page 153 of the “Account of Principles and Methods, &c.”—[ED. J.I.A.]
page 86 note * See Dr.Sprague, 's investigation of this point (J.I.A., xxvi, 109).—[ED. J.I.A.]Google Scholar
page 88 note * The paragraph given above within square brackets is inserted as an Addendum, at Mr. Lidstone's request.—[ED. J.I.A.]
page 92 note *
. The single operation 
gives the value of u
½ obtained by drawing a curve of the third order through u
–1, u
0, u
1, and u
2. The formula will be found in a paper on Graduation contributed by Herr Altenburger in 1905 to the Proceedings of the Austro-Hungarian Insurance Offices' Association (noted under “Additions to the Library” on page 414, of vol. xxxix of the Journal).—[ED. J.I.A.]