Kybernetika 30 no. 3, 279-288, 1994

Stirling distributions and Stirling numbers of the second kind. Computational problems in statistics

François Hennecart


The left-truncated generalized Poisson distribution belongs to the family of the modified power series distributions. Using sufficiency and completeness of $\sum X_i$ ($\min X_i,\sum X_i$)) respectively, when the truncation point is known (resp. unknown), the minimum variance unbiased (M.V.U) estimator for certain functions of the parameter $\theta$ (resp. $\theta,r$) involved in these distributions can be obtained (see Charalambides [3, 4], Jani [10], Voinov-Nikulin [23]). These distributions, as well as the corresponding M.V.U estimators, was expressed in terms of the modified Stirling numbers of the second kind (S.N.S.K). In this paper we give some ways to compute these numbers: first we summarize some usual and less standard identities or relations affecting the S.N.S.K. Some basic properties are given and discussed in view of calculation. Then, by generalizing asymptotic estimates of the usual S.N.S.K, we give and discuss alternative ways to compute the modified S.N.S.K.


62F10, 62E15, 60C05, 05A19, 11B37, 11B73, 11Z05