Discrete distributions for use in twin studies

We describe three discrete distributions motivated by the study of longevity in twins. All individuals both living and dead at a given age are randomly paired. The univariate distribution models the number of these pairs where both individuals are alive. If there is a positive association to longevity in twins then we would expect to see a large number of twin pairs both alive at older ages relative to the number of living individuals. We obtain Poisson and normal approximations to the exact distribution. Multivariate distributions are developed to allow for simultaneous and conditional inference at different ages. An odds-ratio parameter model provides a measure of association of longevity within twin pairs. These models indicate an excessive number of identical twin pairs both alive after age 60 in a cohort of twins born 1870-80 in Denmark.