The most common control chart used to monitor count data is based on Poisson distribution, which presents a strong restriction: The mean is equal to the variance. To deal with under- or overdispersion, control charts considering other count distributions as Negative Binomial (NB) distribution, hyper-Poisson, generalized Poisson distribution (GPD), Conway–Maxwell–Poisson (COM-Poisson), Poisson–Lindley, new generalized Poisson–Lindley (NGPL) have been developed and can be found in the literature. In this paper we also present a Shewhart control chart to monitor count data developed on Touchard distribution, which is a three-parameter extension of the Poisson distribution (Poisson distribution is a particular case) and in the family of weighted Poisson models. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. Consequences in terms of speed to signal departures of stability of the parameters are obtained when incorrect control limits based on non-Touchard distribution (like Poisson, NB or COM-Poisson) are used to monitor count data generated by a Touchard distribution. Numerical examples illustrate the current proposal.

中文翻译：

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使用基于 Touchard 模型的 Shewhart 控制图监控计数数据

用于监控计数数据的最常用控制图是基于泊松分布的，它具有很强的限制：均值等于方差。为了处理分散不足或过度分散，控制图考虑了其他计数分布，如负二项式 (NB) 分布、超泊松、广义泊松分布 (GPD)、康威-麦克斯韦-泊松 (COM-泊松)、泊松-林德利、新的广义Poisson-Lindley (NGPL) 已经被开发并且可以在文献中找到。在本文中，我们还提供了一个 Shewhart 控制图来监控在 Touchard 分布上开发的计数数据，它是 Poisson 分布的三参数扩展（Poisson 分布是一个特殊情况）并且属于加权 Poisson 模型系列。它的归一化常数与 Touchard 多项式有关，因此这个模型的名字。它是一种灵活的分布，可以解释非泊松计数数据中经常发现的分散不足或过度分散以及零点的集中。当基于非 Touchard 分布（如 Poisson、NB 或 COM-Poisson）的不正确的控制限制用于监控由 Touchard 分布生成的计数数据时，获得了速度对参数稳定性偏离的信号的后果。数字示例说明了当前的建议。NB 或 COM-Poisson）用于监控由 Touchard 分布生成的计数数据。数字示例说明了当前的建议。NB 或 COM-Poisson）用于监控由 Touchard 分布生成的计数数据。数字示例说明了当前的建议。