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- W4289743169 abstract "Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $E(z^{X_a})=f(az)/f(a)$. Assume that $Pr(X_aleq k)$ has the form $c_kint_a ^{infty}u^kmu(du)$ for some number $c_k$ and some positive measure $mu,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Moebius functions preserve the cross ratio, textsc{Keywords:} Binomial, Poisson and negative binomial distributions. Gaussian and Gamma distributions. Moebius transforms. Cross ratio." @default.
- W4289743169 created "2022-08-04" @default.
- W4289743169 creator A5008273405 @default.
- W4289743169 date "2018-07-30" @default.
- W4289743169 modified "2023-09-29" @default.
- W4289743169 title "Cumulative distribution functions for the five simplest natural exponential families" @default.
- W4289743169 doi "https://doi.org/10.48550/arxiv.1807.11260" @default.
- W4289743169 hasPublicationYear "2018" @default.
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