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Keeping Abreast of Mortality Change - page 13 / 21

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Appendix 1. Median, percentiles, means and standard deviations according to the Gompertz distribution and the Gompertz curve of deaths.

Pollard and Valkovics (1992) obtained the exact moments and percentiles for the Gompertz distribution with hazard function (1) over the range (-, ). Their formulae may be re- arranged as follows in terms of the mode m and ageing constant k:

mean = m - / k = m - 0.5772157 / k ;

(A1.1)

standard deviation = (/6) / k = 1.2825498 / k ;

(A1.2)

100p percentile = m + [ln(-ln(1-p))] / k ;

(A1.3)

(is the Euler constant). Comparison of (A1.1) and (A1.3) reveals that the mean should correspond to the 43.0% point of the distribution, whilst (A1.3) reveals that the mode must lie at the 63.2% point of the distribution.

Pollard (1991) showed that in respect of a Gompertz life aged x, the following was true of the

time T to death:

E(ekT) = 1 + k / x ;

(A1.4)

standard deviation of ekT

=

k / x .

(A1.5)

100p percentile of T distribution = ln[1 - ek(m-x)

ln(1-p)] .

(A1.6)

Using the standard statistical formulae for the expectation and variance of a random variable under a non-linear transformation, he was able to deduce from (A1.4) and (A1.5) formulae for the expectation and standard deviation of T, which may be re-arranged as:

ex = E(T) [ln(1 + ek(m-x)) - 1/2 (1 + ek(x-m))-2] / k ;

(A1.7)

standard deviation of T [k(1 + ek(x-m))]-1 .

(A1.8)

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