Simple and useful statistical control charts for monitoring count data

• Published in: Quality and reliability engineering international Vol. 37; no. 2; pp. 541 - 566
• Main Authors: Bourguignon, Marcelo, Medeiros, Rodrigo M.R., Fernandes, Fidel Henrique, Lee Ho, Linda
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 01-03-2021
• Subjects: asymptotic control limits; Asymptotic properties; BerG distribution; Control charts; Control limits; count data; Decision theory; Distribution functions; Mean; Monitoring; overdispersion; Parameters; Poisson distribution; Probability distribution functions; Random variables; Statistical analysis; underdispersion
• ISSN: 0748-8017; 1099-1638
• DOI: 10.1002/qre.2748

This paper studies control charts based on the BerG (which is a sum of Bernoulli and geometric random variables) process to deal with the cases of equidispersion, overdispersion, underdispersion, or zero inflation (or deflation). Its probability distribution function can be expressed in terms of the mean parameter and its cumulative distribution has a closed form, thus the construction of an
X¯ control chart to monitor the mean can be made easily. Additionally, we call attention that the asymptotic control limits for
X¯ control chart by central limit theorem (CLT) may lead to a serious erroneous decision. We present guidelines for practitioners about the minimum sample size needed to match out‐of‐control average run length (ARL1) with the exact and asymptotic control limits in function of the shape parameter after an extensive simulation study. The proposed schemes are applied to monitoring the BerG mean parameter.