New MLE algorithm and extended negative binomial parameterization that includes Poisson as a limit case, with proof that Poisson data yields consistent recovery of the true Poisson parameters.
According to Theorem 5.13.1 in Fisz (1963),f NB(ν,p(ν))(k)→f λ(k) asν→ ∞for each k, wherep(ν) = ν ν+λ
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From Poisson Observations to Fitted Negative Binomial Distribution
New MLE algorithm and extended negative binomial parameterization that includes Poisson as a limit case, with proof that Poisson data yields consistent recovery of the true Poisson parameters.