Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy

Abstract We give simple upper and lower bounds on life expectancy. In a life-table population, if e(0) is the life expectancy at birth, M is the median length of life, and e(M) is the expected remaining life at age M, then (M + e(M))/2 ≤ e(0) ≤ M + e(M)/2. In general, for any age x, if e(x) is the expected remaining life at age x, and l(x) is the fraction of a cohort surviving to age x at least, then (x + e(x)) . l(x) ≤ e(0) ≤ x + l(x) . e(x). For any two ages 0 ≤ w ≤ x ≤ ω, (x - w + e(x)) . l(x)/l(w) ≤ e(w) ≤ x - w + e(x) . l(x)/l(w). These inequalities give bounds on e(0) even without detailed knowledge of the course of mortality prior to age x, provided l(x) can be estimated. Such bounds could be useful for estimating life expectancy when the input of eggs or neonates can be estimated but mortality cannot be observed before late juvenile or early adult ages. (ProQuest: ... denotes formulae omitted.) 1. Relationship The life table l(x), constant in time, with continuous age x, is the proportion of a cohort (whether a birth cohort or a synthetic period cohort) that survives to age x or longer. The maximum possible age ω may be finite or infinite. By definition, l(0) = 1 and l(ω) = 0 and l(x) is non-increasing with increasing x, and therefore is nonnegative since l(!) = 0. Assume l(x) is a continuous function of x, 0 ≤ x ≤ ω. The complete expectation of life at age x, e(x), is the average number of years remaining to be lived by those who have attained age x. We show here that e(0), life expectancy at birth, satisfies, for every age x, (1) (x + e(x)) . l(x) ≤ e(0) ≤ x + l(x) . e(x). More generally, for any two ages 0 ≤ w ≤ x ≤ ω, (2) ... When w = 0, (2) reduces to (1). All these inequalities reduce to equality when x = 0 in (1) or x = w in (2) and are strict inequalities if x > 0 or x > w and l(x) is strictly decreasing with increasing x. Subtracting the lower bound in (1) from the upper bound in (1) gives the difference (x - w)(1 - l(x)=l(w)). Both factors in this expression increase with the difference in ages x-w, so the closer the age difference x-w is to 0, the closer the lower and upper bounds are to e(w). Also, the higher the probability l(x)=l(w) of survival from age w to x, the closer the lower and upper bounds are to e(w). The inequalities take a particularly simple form when w = 0 and x = M, the median length of life. Then l(M) = 1=2 and (3) ... Similarly, when w = 0 and x = U, the upper quartile of length of life, then l(U) = 1=4 and (4) ... Also, from (2), (5) ... As a consequence of (1), (6) sup((x + e(x)) . l(x)j0 ≤ x ≤ ω) ≤ e(0) ≤ inf(x + l(x) . e(x)j0 ≤ x ≤ ω): These inequalities make it possible to estimate bounds for e(0) without detailed knowledge of the course of mortality prior to age x, provided that l(x) can be estimated. Such bounds could be useful in estimating life expectancy when the input of eggs or neonates can be estimated but mortality cannot be observed before late juvenile or early adult ages. The lower the mortality before late juvenile or early adult ages, the more closely the bounds will bracket e(0). 2. Proof of (2) A standard formula (Keyfitz 1968:6) for life expectancy at any age w, 0 ≤ w ≤ ω, is (7) ... Therefore, for any ages w ... Since l(x) is non-increasing as x increases, (8) ... …