A generalized statistical control chart for over‐ or under‐dispersed data

Abstract

The Poisson distribution is a popular distribution used to describe count information, from which control charts involving count data have been established. Several works recognize the need for a generalized control chart to allow for data over-dispersion; however, analogous arguments can also be made to account for potential underdispersion. The Conway-Maxwell-Poisson (COM-Poisson) distribution is a general count distribution that relaxes the equi-dispersion assumption of the Poisson distribution, and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. Accordingly, a flexible control chart is developed that encompasses the classical Shewart charts based on the Poisson, Bernoulli (or binomial), and geometric (or negative binomial) distributions.

A flexible distribution class for count data

“…In actuality, it is not necessarily straightforward to determine if observed dispersion is true or "apparent". In a regression setting, for example, dispersion is measured via conditional means and variances, and exploratory data analysis may not detect the true complexity of the data (Sellers and Shmueli 2013). Under such circumstances, the sCMP class can aid with detecting dispersion when a more sophisticated approach is required.…”

“…While we estimate the standard errors of the parameter estimates via the approximate information matrix as described in Section 4, the sampling distributions associated with λ and ν are known to possess skewness (Sellers and Shmueli 2013). Thus, an alternative approach is non-parametric bootstrapping.…”

“…The CMP distribution has quickly grown in popularity because of its ability to model count data in a flexible manner. Methodological developments are vast, including works in distribution theory (Sellers 2012;Sellers and Shmueli 2013;Borges et al 2014), regression analysis (Sellers and Shmueli 2009;2010;Sellers and Raim 2016), control chart Table 1 Well-known distributions associated with the Conway-Maxwell-Poisson (CMP) distribution for special cases of λ and ν…”

On the Modelling and Monitoring of General Inflated Poisson Processes

“…In the current subsection, inspired by the work of Saghir et al . (see also Sellers), we will use the number of surface defects that has been counted on n = 25 rectangular steel plates (Montgomery, p. 338) in order to explore the fitting of various distributional models in a data set that contains some level of dispersion. The sample mean and variance of the data are and s 2 =4.49, with variance/mean ratio equal to 1.90.…”

Attribute Control Charts with Optimal Limits

“…Some studies worked on the need for a general control chart for attribute data. Sellers developed control charts for Poisson, geometric, and binomial distributions. Shore introduced a general approach to attribute control charts that consider mean, standard deviation, and skewness of the underlying distribution.…”