Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy - Joel E. Cohen (Demographic Research, 2011)
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) 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 where the input of eggs or neonates can be estimated but mortality cannot be observed before late juvenile or early adult ages.