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An Approximate Law of Survivorship and Other Notes on the Use of Frequency Curves in Actuarial Statistics

Published online by Cambridge University Press:  18 August 2016

W. Palin Elderton
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Extract

LET EX be the exposed to risk at age x; F (x, 0) be the ordinate at x of a frequency curve used instead of Ex : the frequency curve is not necessarily a graduation of Ex. but may be and in some circumstances will be;

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 65 , Issue 1 , March 1934 , pp. 1 - 36
Copyright
Copyright © Institute and Faculty of Actuaries 1934

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References

page 3 note * All references are to Pearson's system of curves, except when otherwise stated.

page 4 note * Biometrika, Vol. 11, p. 260.

In the same volume of Biometrika, p. 503, an example was given in which an arbitrary normal curve was taken for the exposed, and the resulting Exqx was graduated by two normal curves having the same standard deviation and one having the same mean as the arbitrary curve. Makeham's law of mortality can thus be fitted. See also Frequency Curves and Correlation, pp. 117–20; Rowland, S. J. and Elderton, W. P., J.I.A. Vol. L, p. 251 Google Scholar; Hardy, G. F., Theory of Construction of Tables of Mortality, etc., Note G (London: C. and E. Layton, 1909).Google Scholar

page 7 note * Transactions of International Congress of Mathematicians at Toronto, 1924, p. 867.

page 8 note * The most hopeful seemed to be (1) calculate from F (x, 0) the exact moments to a large number of decimal points, (2) take F (x, 0) qx and calculate moments exactly, (3) find the moments for F (x, 0) px by subtraction of (2) from (1) and fit therefrom. This would be further improved if we could fit F (x, 0) qx directly to F(x, 0) − F(x, 1).

page 8 note † Based on New Entrants, with profits, whole life, 1863–93 experience.

page 8 note ‡ Based on Existing, with profits, whole life, 1863–93 experience.

page 15 note * Charlier has also suggested a C series but it seems hardly likely to be helpful in this type of problem.

page 15 note † An example is given and discussed in Appendix VII.

page 21 note * Vol. I, Ch. 1.

page 21 note † This line of thought has no connection with the theory that mortality tables can be obtained from deaths alone by dissecting the mortality into causes of death. (See comment in J.I.A. Vol. LIV, pp. 205 and 206.)

page 22 note * Is it a possible form ? If k is large relatively to nP80 it would appear ridiculous at nP20 and there comes an age above 80 where one table continues and the other ends.

page 22 note † This was in connection with valuation in a single group and English actuaries interested in this kind of problem may be referred to Otto Draminsky's paper in Skandinavisk Aktuarietidskrift, 1921, p. 339, “Calculation of Reserves by Lidstone's and Givskov's Methods.” P. N. H. Givskov in Praemiereserve, Copenhagen, 1914, gives detailed calculations. Mr Givskov tells me that he has made some alterations in his method since Dr Draminsky's paper and that Professor Marchand has applied the method in Switzerland. Givskov's methods are quite independent of the law of survivorship discussed here and are somewhat similar in idea to Lidstone's Z method.