A new discrete distribution for count data called extended biparametric Waring ($EBW$) distribution is developed. Its name is related to the fact that, in a specific configuration of its parameters, it can be seen as a biparametric version of the univariate generalized Waring ($UGW$) distribution, a well-known model for the variance decomposition into three components: randomness, liability and proneness. Unlike the $UGW$ distribution, the $EBW$ can model both overdispersed and underdispersed data sets. In fact, the $EBW$ distribution is a particular case of a $UWG$ distribution when its first parameter is positive; otherwise, it is a particular case of a Complex Triparametric Pearson ($CTP$) distribution. Hence, this new model inherits most of their properties and, moreover, it helps to solve the identification problem in the variance components of the $UGW$ model. We compare the $EBW$ with the $UGW$ by a simulation study, but also with other over and underdispersed distributions through the Kullback-Leibler divergence. Additionally, we have carried out a simulation study in order to analyse the properties of the maximum likelihood parameter estimates. Finally, some application examples are included which show that the proposed model provides similar or even better results than other models, but with fewer parameters.

中文翻译：

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Waring分布的过度和欠分散双参数扩展

计数数据的新离散分布称为扩展双参数Waring（$Ë乙\mathrm{w\; ^}$）开发发行。它的名称与以下事实有关：在其参数的特定配置中，可以将其视为单变量广义Waring（$üG\mathrm{w\; ^}$）分布，将方差分解为三个部分的著名模型：随机性，责任感和倾向性。不像$üG\mathrm{w\; ^}$ 分布， $Ë乙\mathrm{w\; ^}$可以对过度分散和欠分散的数据集建模。实际上，$Ë乙\mathrm{w\; ^}$ 分布是一个特殊的例子 $ü\mathrm{w\; ^}G$当其第一个参数为正时分布；否则，这是复三参数皮尔逊（$CŤP$）分发。因此，该新模型继承了它们的大多数属性，而且有助于解决模型的方差成分中的识别问题。$üG\mathrm{w\; ^}$模型。我们比较$Ë乙\mathrm{w\; ^}$ 与 $üG\mathrm{w\; ^}$通过模拟研究，还可以通过Kullback-Leibler发散与其他过度分散和欠分散的分布进行比较。此外，我们进行了仿真研究，以分析最大似然参数估计值的属性。最后，包括一些应用示例，这些应用示例表明所提出的模型提供了比其他模型相似甚至更好的结果，但是参数更少。