# A Bayesian Control Chart for Monitoring Process Variance
## 1 Introduction
[PERSON] [26] proposed a Bayesian approach to obtaining control charts, where a predictive distribution based on a Bayesian approach was used to derive the control limits. Furthermore, based on the predictive distributions, the control charts for generalized variance, variance, and coefficient of variation have been done [27; 28]. It is worth mentioning that [PERSON] [29] developed exponentially weighted moving average (EWMA) control charts for mean and variance of a normal distribution with the Bayesian approach, where the posterior predictive distribution has incorporated the parameter uncertainty. [PERSON] [30] used informative and non-informative priors for updating the process mean and obtained the X-bar chart based on posterior control limits. [PERSON] et al. [31] proposed a new design for a control chart using a modified EWMA statistic under the assumption that the quality characteristic of interest follows the normal distribution. Based on the framework of [PERSON] et al. [31], [PERSON] et al. [32] proposed a modified EWMA control chart for monitoring the process variance. Recently, [PERSON] and [PERSON] [33] proposed a new Bayesian Modified-EWMA chart for the monitoring of the location parameter in a process, and applied the proposed scheme to the mechanical and sport industry. From the reviews of the above literatures, we found that most of the Bayesian control charts were discussed under normality, and almost no literature addresses Bayesian control charts under distribution-free processes. This study will focus on the Phase II monitoring. Based on the framework of [PERSON] and [PERSON] [23], a Bayesian procedure is applied to obtain the control limits for monitoring the process variance without the assumption of a normal process distribution, a distribution of known form, or a distribution with fixed skewness. We explore the sampling properties of the new monitoring statistic and find that it is suitable for monitoring the time-varying process. The average run lengths (ARLs) of the proposed Bayesian control chart are calculated and show that the chart performs well. The simulation studies for a normal process, exponential process, and the mixed process of normal and exponential distribution prove that our chart can quickly detect any shift of process variance. Finally, a numerical example of service time is used to illustrate the application of the proposed Bayesian EWMA variance chart and confirm the performance of process control. The article is organized as follows. In Section 2, we describe the preliminary settings of our control scheme and the distribution of a new statistic derived from the Bayesian approach. Then, the construction of a newly proposed Bayesian variance chart and its performance are discussed in Section 3. In Section 4, we propose a Bayesian exponentially weighted moving average (EWMA) variance chart. In Section 5, a simulation study for a normal process, exponential process, and the mixed process of normal and exponential distributions is performed to test the detection performance of the proposed Bayesian EWMA variance chart. In Section 6, a numerical example of a service system in a bank branch is used to illustrate how to construct the proposed variance chart. Finally, we summarize the findings and provide recommendations in Section 7.