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Stirling distributions and Stirling numbers of the second kind. Computational problems in statistics. (English) Zbl 0810.62029

Summary: 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 (MVU) estimator for certain functions of the parameter \(\theta\) (resp. \(\theta\), \(r\)) involved in these distributions can be obtained.
These distributions, as well as the corresponding MVU estimators, were expressed in terms of the modified Stirling numbers of the second kind (SNSK). We give some ways to compute these numbers: first we summarize some usual and less standard identities or relations affecting the SNSK. Some basic properties are given and discussed in view of calculation. Then, by generalizing asymptotic estimates of the usual SNSK, we give and discuss alternative ways to compute the modified SNSK.

MSC:

62F10 Point estimation
11B37 Recurrences
62E15 Exact distribution theory in statistics
05A19 Combinatorial identities, bijective combinatorics
11Z05 Miscellaneous applications of number theory

References:

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