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Existence and uniqueness of the maximum likelihood estimator for the two-parameter negative binomial distribution

Author

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  • Aragón, Jorge
  • Eberly, David
  • Eberly, Shelly

Abstract

Given a sample with mean x and second moment s2, Anscombe in 1950 conjectured that the maximum likelihood equations for the two-parameter negative binomial distribution have a unique solution if and only if s2 > x. We give a proof of his conjecture.

Suggested Citation

  • Aragón, Jorge & Eberly, David & Eberly, Shelly, 1992. "Existence and uniqueness of the maximum likelihood estimator for the two-parameter negative binomial distribution," Statistics & Probability Letters, Elsevier, vol. 15(5), pages 375-379, December.
    • Handle: RePEc:eee:stapro:v:15:y:1992:i:5:p:375-379
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    Cited by:

    1. Katiane S. Conceição & Marinho G. Andrade & Francisco Louzada & Nalini Ravishanker, 2022. "Characterizations and generalizations of the negative binomial distribution," Computational Statistics, Springer, vol. 37(3), pages 1255-1286, July.
    2. Shilane David & Evans Steven N & Hubbard Alan E., 2010. "Confidence Intervals for Negative Binomial Random Variables of High Dispersion," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-41, March.
    3. Vera Hofer & Johannes Leitner, 2012. "A bivariate Sarmanov regression model for count data with generalised Poisson marginals," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(12), pages 2599-2617, August.
    4. Ferreri, Carlo, 1997. "On the ML-estimator of the positive and negative two-parameter binomial distribution," Statistics & Probability Letters, Elsevier, vol. 33(2), pages 129-134, April.
    5. Wang, Yining, 1996. "Estimation problems for the two-parameter negative binomial distribution," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 113-114, February.

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