Determining the lifetime density function using a continuous approach

Objective: To apply a continuous hazard function approach to calculate the lifetime density function (LDF) at any age, and to compare the life expectancies derived from the LDF with those obtained with standard life table (SLT) methods. Methods: Age-specific mortality rates were modeled through a continuous hazard function. To construct the cumulative hazard function, appropriate integration limits were considered as continuous random variables. The LDF at any age was defined on the basis of the elemental relationships with the cumulative hazard function. Life expectancies were calculated for a particular set of mortality data using the SLT approach and the expectancy of the LDF defined. Applications and comparisons: The proposed approach was applied using mortality data from the 2001 census of Catalonia (Spain). A Gompertz function was used to model the observed age-specific mortality rates, which fitted the observed data closely. The LDF and the life expectancy, median and standard deviation of the LDF were derived using mathematical software. All differences, in percentages, between the life expectancies obtained from the two methods were 1.1% or less. Conclusions: The LDF gives a wider interpretation of life duration, by extending a deterministic value like life expectancy to a fully informative measure like the LDF.