On the existence of maximum likelihood estimates for the parameters of the Conway-Maxwell-Poisson distribution

Pub Date : 2023-01-01 DOI:10.30757/alea.v20-20
S. Bedbur, U. Kamps, A. Imm
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Abstract

. As a well-known and important extension of the common Poisson model with an additional parameter, Conway-Maxwell-Poisson (CMP) distributions allow for describing under-and overdispersion in discrete data. Constituting a two-parameter exponential family, CMP distributions possess useful structural and statistical properties. However, the exponential family is not steep and maximum likelihood estimation may fail even for non-trivial data sets, which is different from the Poisson case, where maximum likelihood estimation only fails if all data outcomes are zero. Conditions are examined for existence and non-existence of maximum likelihood estimates in the full family as well as in subfamilies of CMP distributions, and several figures illustrate the problem.
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康威-麦克斯韦-泊松分布参数的极大似然估计的存在性
。康威-麦克斯韦-泊松分布(Conway-Maxwell-Poisson, CMP)是普通泊松模型的一个众所周知的重要扩展,它增加了一个参数,允许描述离散数据中的欠散和过散。CMP分布构成一个双参数指数族,具有有用的结构和统计性质。然而,指数族并不陡峭,即使对于非平凡数据集,最大似然估计也可能失败,这与泊松情况不同,在泊松情况下,最大似然估计只有在所有数据结果为零时才会失败。考察了在CMP分布的全族和亚族中存在和不存在最大似然估计的条件,并用几个图说明了这个问题。
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