X hits on this document

PDF document

Keeping Abreast of Mortality Change - page 14 / 21

51 views

0 shares

0 downloads

0 comments

14 / 21

Appendix 2. The effect on the complete expectation of life of changing the ageing parameter in the Gompertz “law”

Consider the function

t

Z = - l n t p x

= x+u

du .

(A2.1)

0

Under Gompertz mortality, this function becomes

t

Z = x eku du .

(A2.2)

0

For given fixed x but varying senescence parameter k therefore

t

dZ/dk = x u eku du ,

(A2.3)

0

which on integration and rearrangement yields

dZ/dk = (k t x+t - x+t

  • +

    x)/k2

.

(A2.4)

W e n o w n o t e t h a t s i n c e t p x = e - Z ,

d ( t p x ) / d k = - t p x d Z / d k .

(A2.5)

Under the Gompertz “law”, therefore,

d ( t p x ) / d k = ( - k t t p x x + t + t p x x + t

- x t p x ) / k 2 .

(A2.6)

Integration of (A2.6) with respect to t from 0 to yields

d(ex)/dk = [1 - (x + k) ex]/k2 .

(A2.7)

Multiplication of (A2.7) by k2 and differentiation with respect to k (keeping x constant) leads to

d2(ex)/dk2 = -[(x+3k) d(ex)/dk + ex] / k2 ,

(A2.8)

a process which may be repeated to obtain higher-order derivatives.

Document info
Document views51
Page views51
Page last viewedFri Dec 09 05:02:27 UTC 2016
Pages21
Paragraphs757
Words5648

Comments