Copula-Based Bivariate ZIP Control Chart for Monitoring Rare Events

Abstract

One difficulty with developing multivariate attribute control charts is the lack of the related joint distribution. So, if it would be possible to generate the joint distribution of two (or more) attribute characteristics, then a bivaraite (or multivariate) attribute control chart can be developed based on Types I and II errors. Copula function is a solution to the matter. In this article, applying the copula function approach, we achieve the joint distribution of two correlated zero inflated Poisson (ZIP) distributions. Then, using this joint distribution, we develop a bivaraite control chart which can be used for monitoring correlated rare events. This copula-based bivariate ZIP control chart is compared with the simultaneous use of two separate univariate ZIP control charts. Based on the average run length (ARL) measure, it is shown that the proposed control chart is much better than the simultaneous use of two separate univariate charts. In addition, a real case study related to the environmental air in a sterilization process is investigated to show the applicability of the developed control chart.

Keywords:

• Control chart
• Copula function
• SPC
• Zero inflated Poisson (ZIP) distribution
• Rare events

Acknowledgments

We would like to thank the Editor and referees for their insightful and detailed comments that greatly improved the article. Also, we thank Miss Samar Omidi Noubijari for her cooperation in revising the text.