论文标题
关于任意分散的Conway-Maxwell-Poisson分布
On arbitrarily underdispersed Conway-Maxwell-Poisson distributions
论文作者
论文摘要
我们表明，当通过其平均值进行参数化时，Conway-Maxwell-Poisson分布可能被任意不足。更确切地说，如果平均$μ$是整数，则限制分布是$μ$的单位概率质量质量。如果平均$μ$不是整数，那么限制分布是两个值$ \floailμ$和$ \ceilμ$的伯诺利，概率等于$μ$的分数零件。在任何一种情况下，限制分布都是任何给定均值的分散分布最少的分布。目前，这是展示此属性的泊松分布的唯一已知的概括。讨论了四个实际含义，每种含义都增加了（均值的）Conway--Maxwell-Poisson分布的说法，应视为默认分散计数的默认模型。我们建议对泊松分布进行所有未来的概括，以针对此属性进行测试。
We show that the Conway--Maxwell--Poisson distribution can be arbitrarily underdispersed when parametrized via its mean. More precisely, if the mean $μ$ is an integer then the limiting distribution is a unit probability mass at $μ$. If the mean $μ$ is not an integer then the limiting distribution is a shifted Bernoulli on the two values $\floorμ$ and $\ceilμ$ with probabilities equal to the fractional parts of $μ$. In either case, the limiting distribution is the most underdispersed discrete distribution possible for any given mean. This is currently the only known generalization of the Poisson distribution exhibiting this property. Four practical implications are discussed, each adding to the claim that the (mean-parametrized) Conway--Maxwell--Poisson distribution should be considered the default model for underdispersed counts. We suggest that all future generalizations of the Poisson distribution be tested against this property.
