Improving estimation of the variance of expectation of life for small populations

It has been pointed out (D Eayres, personal communication) that Chiang’s formula1 for estimation of the standard error of the expectation of life (which allows for the fact that mortality rates increase with age within each age band—clearly relevant with abridged life tables and advancing age) can be improved by combining it with the fact that the final age band is open-ended so that its width is a random variable and therefore contributes to the standard error of expectation of life.2 However, in simulations Eayres notes (personal communication) that there is still bias in the estimation of expectation of life, especially with small populations. The purpose of this communication is to suggest an improvement to reduce the effect of this bias. Errors in estimation of the population may also cause spurious variation in e^o_x between populations, but these are not considered here.

METHODS

In Silcocks et al2 the expectation of life (e^o_x) is expressed in terms of functions of the age specific mortality rates. The (theoretically) open-ended final age band contributes to the point estimate of e^o_x and its variance by assigning to the final age band a notional fixed width, which in turn is a function of that age band’s mortality rate.

Assuming …