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On Mr. Gompertz's Method for the Adjustment of Tables of Mortality
Published online by Cambridge University Press: 18 August 2016
Extract
When an observation has been made for the purpose of determining the mortality which has prevailed during a specified time in the community observed upon, it is usually found that the series of probabilities of living a year, at the various ages thence deduced, is very far from conforming to our ideas of what an exponent of the law of mortality ought to be. Instead of a series exhibiting throughout its extent a gradual and progressive change of value in passing from each term to the next, we have one showing but a faint approach to the desired regularity. This is just what, from the circumstances in which such observations are usually made, we ought to expect. It is not often that the community subjected to observation is sufficiently large, or that the period during which it is so subjected is sufficiently extended, to furnish us with such a series of average results as can alone fitly figure forth the law of mortality.
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- Copyright © Institute and Faculty of Actuaries 1857
References
page 122 note * The word in Mr. Gompertz's Memoir is, ‘portions’; but this is most likely a misprint.
page 123 note * A familiar example of a function of this form is that which denotes the present value of a sum a, due at the end of the time x, namely avx. This function in the time h loses avx - avx+h = avx(1 - vh) which is to the initial value, avx , in the ratio 1 – vh . 1, a ratio independent of x.
page 124 note † Thus the amount of a sum a, in the time x, is reciprocally proportional to its present value for the same time; and both are exponentials.
page 125 note * Friendly Societies Report, 1827, pp. 62, 64.
page 126 note * The first and second differences are negative, and negative numbers have no arithmetical logarithms. Nevertheless, the logarithms belonging to such numbers, considered as positive, admit of being used for purposes of multiplication and division.
page 127 note * Mr. Finlaison describes his method in his Report on Life Annuities, 31st March, 1829. The description seems plain enough, and yet I confess that, on applying it to the data in his 13th and 20th Observations (which are those on which the present Government Annuities are founded), I have not been able to bring out his results. Mine differ widely from his. I have succeeded readily in verifying Mr. Galloway's application of the method, so that I do not think there is room to suppose that I have misapprehended it. There are also, in the tables referred to, several discrepancies between the values in the column of adjusted probabilities and in that headed “Law of Mortality”, but it does not seem worth while pointing them out.