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

Given a sample with mean x̄ and second moment s 2, Anscombe in 1950 conjectured that the maximum likelihood equations for the two-parameter negative binomial distribution have a unique solution if and only if s 2 > x̄. We give a proof of his conjecture.

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Bibliographic Details
Published in:Statistics & probability letters 1992-12, Vol.15 (5), p.375-379
Main Authors: Aragón, Jorge, Eberly, David, Eberly, Shelly
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematics
Maximum likelihood estimator
Maximum likelihood estimator negative binomial distribution Newton's method
negative binomial distribution
Newton's method
Parametric inference
Probability and statistics
Sciences and techniques of general use
Statistics
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Summary:Given a sample with mean x̄ and second moment s 2, Anscombe in 1950 conjectured that the maximum likelihood equations for the two-parameter negative binomial distribution have a unique solution if and only if s 2 > x̄. We give a proof of his conjecture.
ISSN:0167-7152
1879-2103
DOI:10.1016/0167-7152(92)90157-Z