AbstractIn this paper the distribution of Z = Σni=1Xi and the joint distribution of Y = min{X1,…,Xn} and Z are obtained in the cases in which X1,…,Xn is a random sample (i) from the left‐truncated generalized logarithmic series distribution in terms of the Stirling numbers of the first kind, (ii) from the left‐truncated generalized Poisson distribution in terms of the Stirling numbers of the second kind and (iii) from the left‐truncated generalized negative binomial in terms of linear combinations of the Stirling numbers of both kinds. These distributions are utilized in order to obtain the minimum variance unbiased estimators of θm, m ≫ 1 when the truncation point r is assumed to be known and of rm and θm when r is unknown.