A two-parameter hazard function to describe age patterns of mortality in ancient Northwestern Europe

When seeking to describe the age patterns of mortality for ancient populations, researchers are often confronted with small sample sizes or with missing data for several age groups. The traditional approach to dealing with these challenges is to smooth or complement such patterns by matching them to a model life table, either directly or through the Brass logit transformation. This procedure requires an appropriate model life table, which may not be available. We propose a hazard model that is both flexible enough to accurately describe an age pattern of mortality in ancient Northwestern Europe and restrictive enough to complement incomplete data. This paper presents a hazard function that contains four free-to-choose parameters. Tested against a large collection of life tables for northwestern European countries from the 17th to the 21st century, the number of free-to-choose parameters is stepwise reduced from four to only two. Compared with the Brass logit transformation with the Princeton Model West as its reference, the presented two-parameter hazard model is shown to fit the abovementioned dataset much better. The mean fitting error is found to be half the size. Moreover, this model is shown to fit a 13th-century mortality age pattern much better. The proposed two-parameter hazard model is capable of fitting a wide range of age patterns of mortality more closely than the traditional approach can. We therefore conclude that the proposed model facilitates the smoothing and the completion of age patterns of mortality in ancient Northwestern Europe even if they deviate substantially from well-documented patterns.