# A Bayesian Control Chart for Monitoring Process Variance
## 3 The Construction of the Shewhart-Type Variance Chart with the Bayesian Approach
### The Parameters of the Prior Distribution of \(p\) and Process Variance Are Known
#### 3.1.2 The Out-of-Control ARL of the Shewhart-Type Bayesian Variance Chart
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Panel A:** & **When** & \(\mathbf{(\alpha_{0},\beta_{0})=}\) & \(\mathbf{(5,10)}\) & & & & & & \\ \hline \(\mathbf{(\alpha_{1},\beta_{1})}\) & \(\mathbf{E_{1}(p)}\) & \(\mathbf{n=2}\) & \(\mathbf{n=3}\) & \(\mathbf{n=5}\) & \(\mathbf{n=10}\) & \(\mathbf{n=15}\) & \(\mathbf{n=20}\) & \(\mathbf{n=25}\) \\ \hline \((6,11)\) & \(0.3529\) & \(\infty\) & \(\infty\) & \(\infty\) & \(212.26\) & \(423.95\) & \(711.36\) & \(401.99\) \\ \((7,10)\) & \(0.4118\) & \(\infty\) & \(\infty\) & \(\infty\) & \(91.49\) & \(155.39\) & \(231.17\) & \(130.77\) \\ \((6,10)\) & \(0.3750\) & \(\infty\) & \(\infty\) & \(\infty\) & \(141.98\) & \(257.23\) & \(399.88\) & \(226.12\) \\ \((5,10)\)a  & \(0.3333\) & \(\infty\) & \(\infty\) & \(\infty\) & \(240.62\) & \(468.98\) & \(765.58\) & \(433.45\) \\ \((5,9)\) & \(0.3571\) & \(\infty\) & \(\infty\) & \(\infty\) & \(153.86\) & \(270.31\) & \(407.36\) & \(231.39\) \\ \((5,8)\) & \(0.3846\) & \(\infty\) & \(\infty\) & \(\infty\) & \(96.21\) & \(152.44\) & \(212.41\) & \(121.84\) \\ \hline \hline
**Panel B:** & **When** & \(\mathbf{(\alpha_{0},\beta_{0})=}\) & \(\mathbf{(3,9)}\) & & & & & \\ \hline \(\mathbf{(\alpha_{1},\beta_{1})}\) & \(\mathbf{E_{1}(p)}\) & \(\mathbf{n=2}\) & \(\mathbf{n=3}\) & \(\mathbf{n=5}\) & \(\mathbf{n=10}\) & \(\mathbf{n=15}\) & \(\mathbf{n=20}\) & \(\mathbf{n=25}\) \\ \hline \((4,10)\) & \(0.2857\) & \(\infty\) & \(\infty\) & \(153.00\) & \(98.96\) & \(270.31\) & \(214.86\) & \(189.35\) \\ \((5,9)\) & \(0.3571\) & \(\infty\) & \(\infty\) & \(68.00\) & \(38.51\) & \(85.33\) & \(65.18\) & \(55.79\) \\ \((4,9)\) & \(0.3077\) & \(\infty\) & \(\infty\) & \(110.50\) & \(66.73\) & \(162.15\) & \(125.90\) & \(109.03\) \\ \((3,9)\) a  & \(0.2500\) & \(\infty\) & \(\infty\) & \(208.00\) & \(136.39\) & \(368.68\) & \(292.32\) & \(256.96\) \\ \((3,8)\) & \(0.2727\) & \(\infty\) & \(\infty\) & \(143.00\) & \(86.90\) & \(206.94\) & \(160.26\) & \(138.47\) \\ \((3,7)\) & \(0.3000\) & \(\infty\) & \(\infty\) & \(95.33\) & \(53.99\) & \(113.11\) & \(86.08\) & \(73.45\) \\ \hline \hline
**Panel C:** & **When** & \(\mathbf{(\alpha_{0},\beta_{0})=}\) & \(\mathbf{(46,100)}\) & & & & & \\ \hline \(\mathbf{(\alpha_{1},\beta_{1})}\) & \(\mathbf{E_{1}(p)}\) & \(\mathbf{n=2}\) & \(\mathbf{n=3}\) & \(\mathbf{n=5}\) & \(\mathbf{n=10}\) & \(\mathbf{n=15}\) & \(\mathbf{n=20}\) & \(\mathbf{n=25}\) \\ \hline \((70,100)\) & \(0.4118\) & \(\infty\) & \(\infty\) & \(77.86\) & \(57.21\) & \(65.44\) & \(29.00\) & \(39.88\) \\ \((60,100)\) & \(0.3750\) & \(\infty\) & \(\infty\) & \(121.93\) & \(104.50\) & \(137.11\) & \(61.39\) & \(94.91\) \\ \((50,100)\) & \(0.3333\) & \(\infty\) & \(\infty\) & \(213.75\) & \(225.07\) & \(354.80\) & \(164.14\) & \(289.74\) \\ \((46,100)\) a  & \(0.3151\) & \(\infty\) & \(\infty\) & \(279.22\) & \(325.57\) & \(562.03\) & \(265.77\) & \(477.10\) \\ \((46,90)\) & \(0.3382\) & \(\infty\) & \(\infty\) & \(196.80\) & \(199.24\) & \(301.99\) & \(138.61\) & \(237.21\) \\ \((46,80)\) & \(0.3651\) & \(\infty\) & \(\infty\) & \(135.10\) & \(118.39\) & \(157.31\) & \(70.67\) & \(109.53\) \\ \((46,70)\) & \(0.3966\) & \(\infty\) & \(\infty\) & \(89.95\) & \(68.11\) & \(79.34\) & \(35.32\) & \(48.98\) \\ \hline \hline
**Panel D:** & **When** & \(\mathbf{(\alpha_{0},\beta_{0})=}\) & \(\mathbf{(32,100)}\) & & & & & \\ \hline \(\mathbf{(\alpha_{1},\beta_{1})}\) & \(\mathbf{E_{1}(p)}\) & \(\mathbf{n=2}\) & \(\mathbf{n=3}\) & \(\mathbf{n=5}\) & \(\mathbf{n=10}\) & \(\mathbf{n=15}\) & \(\mathbf{n=20}\) & \(\mathbf{n=25}\) \\ \hlineOn the other hand, when there is no information about the parameters of a prior distribution of \(p\), a non-informative prior distribution, \(Beta(1,\,1)\), is suggested initially. Then, the preliminary sample data can be used to obtain the posterior distribution of \(p\):  
\[p|n,T,m_{1},\ \ldots,\ m_{T}\ \sim\ Beta\left(1+\sum_{i=1}^{T}m_{t},\ 1+nT\sum_{i=1}^{T}m_{t}\right),\]  
where \(m_{t}\) represents the number of \(Y_{j}\)'s \(>\vartheta^{2}\) at the \(t\)th sampling period.