On Ewens' equivalence theorem for ascertainment sampling schemes

The usual likelihood formulations for segregation analysis of a genetic trait ignore both the at-risk but unobservable families and the demographic structure of the surrounding population. Families are not ascertained if, by chance, they have no affected members or if the affected members are not ascertained. Ewens has shown that likelihoods which take into explicit account both unobservable families and demographic parameters lead to the same maximum likelihood estimates of segregation and ascertainment parameters as the usual likelihoods. This paper provides an alternative proof of Ewens' theorem based on the Poisson distribution and simple continuous optimization techniques.