Karien Adamski, Schalk W. Human, Andriette Bekker

Practitioners in the quality control environment require methods to monitor a process from the start of the production. Quesenberry (1991) presented Q-charts assuming that the observations from each sample are independent and identically distributed normal random variables. Human and Chakraborti (2010) proposed a Q-chart design for monitoring the process average when the measurements are from an exponential distribution and the parameter of the distribution is unknown. To gain more insight into the performance of a control chart, one needs to consider the run-length distribution of the proposed chart. To develop exact expressions for the probabilities of run-lengths the joint distribution of the charting statistic is needed.

This gives rise to a new distribution that can be regarded as a generalized multivariate beta distribution. This generalized multivariate beta distribution is constructed from independent chi-squared random variables using the variables-in-common (or trivariate reduction) technique. The form of the construction of these random variables and their dependence structure originated from a problem identified in Statistical Process Control (SPC).

An overview of the problem statement is given and the newly developed generalized multivariate beta distribution is proposed.

**Bibliography:**

Human SW, Chakraborti S (2010) Q charts for the exponential distribution. JSM 2010 Proceedings, Section on Quality and Productivity, Vancouver, British Columbia, Canada

Quesenberry CP (1991) SPC Q Charts for start-up processes and short or long runs. Journal of Quality Technology, 23(3), 213 – 224.

**Keywords:** Multivariate Beta; Chi-squared ratios; Variables-in-common technique; Shape analysis

**Biography:** Karien Adamski is a lecturer at the University of Pretoria in South Africa. This research forms part of her PhD studies under the supervision of Prof. Bekker and Dr. Human.