*1.1. Problem Contextualisation*

The monitoring of the variance of independent and identically distributed (i.i.d) normal random variables over time by taking successive, independent samples of measurements over time remains an interesting and valuable research consideration within quality control environment. In this case, when the variance *σ*<sup>2</sup> changes to *σ*<sup>2</sup> <sup>1</sup> = *λσ*<sup>2</sup> for some *λ* = 1, the practitioner needs to investigate the scope of such a change (a value of *λ*), and ideally, the position within the successive measurements where such a change could've taken place. Suppose that *Xij* are i.i.d. *<sup>N</sup>*(*μ*, *<sup>σ</sup>*2), *<sup>i</sup>* = 0, 1, 2, ··· , *<sup>κ</sup>* − 1 and *Xij* <sup>∼</sup> i.i.d. *<sup>N</sup> μ*, *σ*<sup>2</sup> <sup>1</sup> = *λσ*<sup>2</sup> , *i* = *κ*, *κ* + 1, ··· , *m* where *j* = 1, 2, ··· , *ni* ≥ 2 and *λ* > 0, as outlined in Figure 1. The values of *κ* and *λ* are assumed to be unknown, but deterministic in nature. The order of these samples is important and cannot be re-ordered; in other words, the samples have a set sequence corresponding to the order in which they were obtained.

Thus, inspired by a practical objective, this paper aims to present a theoretically motivated framework to


**Citation:** Human, S.W.; Bekker, A.; Ferreira, J.T., Mijburgh, P.A. A Bivariate Beta from Gamma Ratios for Determining a Potential Variance Change Point: Inspired from a Process Control Scenario. *Math. Comput. Appl.* **2022**, *27*, 61. https:// doi.org/10.3390/mca27040061

Academic Editor: Sandra Ferreira

Received: 2 June 2022 Accepted: 14 July 2022 Published: 16 July 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

5. determine whether *λ* = 1, and if this is indeed the case, to determine *κ*, the location of where the shift in the variance occurred.

Therefore, from sample *κ* onwards, the process would be considered out-of-control. Note that it is assumed that the shift occurs between two samples.

**Figure 1.** Process shift.

Assume that both the process mean (*μ*) and variance *σ*2 are unknown, and that they are estimated by their respective minimum variance unbiased estimators (MVUE), given by:

$$\mathcal{R}\_i \quad = \begin{array}{c} \frac{\sum\_{j=1}^{n\_i} X\_{ij}}{n\_i}, i = 0, 1, 2, \cdots, m\_i \end{array} \tag{1}$$

$$S\_i^2 \quad = \frac{1}{n\_i - 1} \sum\_{j=1}^{n\_i} \left( X\_{ij} - \vec{X}\_i \right)^2, i = 0, 1, 2, \cdots, m,\tag{2}$$

where *X*¯*<sup>i</sup>* and *S*<sup>2</sup> *<sup>i</sup>* denote the mean and variance of sample *i* respectively. Some particular notes on Equation (1) and (2) include:


The problem of determining if a shift in the process variance has occurred can be divided into two stages, namely before the potential shift and after, as indicated below (*Gamma*(·, ·) denotes the usual gamma distribution with suitable shape and scale parameters [1]).

