*Article* **A Bivariate Beta from Gamma Ratios for Determining a Potential Variance Change Point: Inspired from a Process Control Scenario**

**Schalk W. Human 1, Andriette Bekker 1,2,\*, Johannes T. Ferreira 1,2 and Philip Albert Mijburgh <sup>1</sup>**


**Abstract:** Within statistical process control (SPC), normality is often assumed as the underlying probabilistic generator where the process variance is assumed equal for all rational subgroups. The parameters of the underlying process are usually assumed to be known—if this is not the case, some challenges arise in the estimation of unknown parameters in the SPC environment especially in the case of few observations. This paper proposes a bivariate beta type distribution to guide the user in the detection of a permanent upward or downward step shift in the process' variance that does not directly rely on parameter estimates, and as such presents itself as an attractive and intuitive approach for not only potentially identifying the magnitude of the shift, but also the position in time where this shift is most likely to occur. Certain statistical properties of this distribution are derived and simulation illustrates the theoretical results. In particular, some insights are gained by comparing the newly proposed model's performance with an existing approach. A multivariate extension is described, and useful relationships between the derived model and other bivariate beta distributions are also included.

**Keywords:** bivariate beta; gamma; hypergeometric function; sequential; shift in process variance
