Le problème d'Anscombe pour les lois binomiales négatives généralisées

Consider a finite sample from a generalized negative-binomial distribution where both (canonical and index) parameters are unknown. This note proves that both the maximum-likelihood estimate and the moment estimate of the index parameter exist if and only if the sample variance is greater than the sample mean. This extends a result for the negative-binomial distribution that had been conjectured by Anscombe (1950) and later shown by Levin and Reeds (1977).