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The distributions of sufficient statistics of truncated generalized logarithmic series, poisson and negative binomial distributions
Author(s) -
Charalambides Ch.A.
Publication year - 1974
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314698
Subject(s) - mathematics , logarithmic distribution , poisson distribution , negative binomial distribution , stirling number , central binomial coefficient , logarithm , binomial coefficient , series (stratigraphy) , statistics , truncation (statistics) , distribution (mathematics) , combinatorics , binomial distribution , mathematical analysis , paleontology , biology
In this paper the distribution of Z = Σ n i=1 X i and the joint distribution of Y = min { X 1 ,…, X n } and Z are obtained in the cases in which X 1 ,…, X n 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 θ m , m ≫ 1 when the truncation point r is assumed to be known and of r m and θ m when r is unknown.