Distributional properties of ratios of gamma random variables in the context of quality control

Abstract

This study emanates from a practical problem in the statistical process control (SPC) environment where the quality of a process is monitored. Speci cally, where the variance of a process is being assessed to be the same for all samples. In the traditional SPC environment the parameters of the underlying manufacturing process are usually assumed to be known. If, however, they are not known, they need to be estimated. Estimating these parameters and using them in control charts has many associated problems, especially when the samples that are used to calculate the estimates contain few data points. This study proposes a new control chart that is used to detect a shift in the process's variance, but that does not directly rely on parameter estimates, and as such overcomes many of these problem. The development of this newly proposed control chart gives rise to a new beta type distribution. An overview of the problem statement as identi ed in the eld of SPC is given and the newly developed beta type distribution is proposed. Some statistical properties of this distribution are studied and the e ect of di erent parameter choices on the shape of the distribution are investigated, with the focus speci cally on the bivariate case. Through simulation, a comparison study is also performed, comparing the newly proposed model with a generalised version of the Q chart model, which was studied in depth by Adamski (2014).