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On some properties of a class of weighted quasi-binomial distributions. (English) Zbl 1084.62011

A random variable \(X\) has a distribution belonging to the quasi-binomial family considered in this paper if \[ P(X=k)=\binom nk\,\frac{(p+k\phi)^{k+s}\,(1-p-k\phi)^{n-k+t}} {B_n(p,1-p-n\phi;s,t;\phi)}, \] where \[ B_n(p,q;s,t;\phi)=\sum_{k=0}^n \binom{n}{k}\,(p+k\phi)^{k+s}\,(q+(n-k)\phi) ^{n-k+t}; \] here \(p\) and \(q\) are non-negative fractions, \(p+q+n\phi=1\); \(-p/n<\phi<(1-p)/n\), and \(s\) and \(t\) are integers. Weighted distributions are also considered. This class contains many distributions in the literature. In the paper, expressions for the \(r\)-th order descending factorial moments, the \(r\)-th order moments, the \(r\)-th order central moments and the negative moments are given.

MSC:

62E15 Exact distribution theory in statistics
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
62F10 Point estimation
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