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# 统计代写|统计推断代写STATISTICAL INFERENCE代写|MTH412 CUSUM Control Charts

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## 统计代写|统计推断代写STATISTICAL INFERENCE代写|CUSUM Control Charts

An example for the parametric CUSUM control chart for individual data follows.
Example 3.4 A Parametric CUSUM Control Chart
Column (a) of Table $3.6$ presents 30 measurements generated from a process that is normally distributed. The first 20 of these observations were drawn at random from a normal distribution with mean $\mu_{0}=10$ and standard deviation $\sigma_{0}=1$. The last 10 observations were drawn from a normal distribution with mean $\mu_{1}=11$ and standard deviation $\sigma_{0}=1$. Thus, $\mu_{1}=\mu_{0}+\delta \sigma_{0}=10+(1)(1)$ so that $\delta=1$. To illustrate the CUSUM chart, suppose that we are interested in detecting an increase (a shift) in the mean of size $1 \sigma_{0}$, which is a medium size shift. Note that, here, we can use $K$ and $k$ interchangeably, since $\sigma_{0}=1$. The same argument holds for $H$ and $h$. Hawkins (1993; Table 1, p. 465) recommends that, for a shift of about $1 \sigma_{0}$ in the process mean, taking $k=0.5$ and $h=4.77$ gives an $A R L_{\mathrm{IC}}=370$. Accordingly, we apply the parametric CUSUM using $k=0.5$ and $h=4.77$.
The CUSUM charting statistics, $C_{i}^{+}$and $C_{i}^{-}$, are given in Columns (b) and (d) of Table 3.6, respectively, whereas the corresponding quantities, $N^{+}$and $N^{-}$are given in Columns (c) and (e), respectively. To illustrate the calculations, consider period 1. The charting statistics for the first period are calculated as follows
\begin{aligned} C_{1}^{+} &=\max \left[0, X_{1}-\left(\mu_{0}+K\right)+C_{0}^{+}\right]=\max [0,9.45-(10+0.5)+0] \ &=\max [0,-1.05]=0 \end{aligned}
and
\begin{aligned} C_{1}^{-} &=\min \left[0, X_{1}-\mu_{0}+K+C_{0}^{-}\right]=\max [0,9.45-10+0.5+0] \ &=\max [0,-0.05]=0 . \end{aligned}

## 统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Variation

Variation is an important aspect of any analysis and thus it is necessary to monitor the process variation or spread and ensure that it is IC. Moreover, as we see in Equation 3.1, the Shewhart control limits for the process mean depend on the process standard deviation. Thus, unless the standard deviation remains IC, the control chart for the mean will not be very informative. So, we need to monitor the variance or the standard deviation using a control chart.

There are several possible statistics that can be used to monitor variation. The most popular choices are the sample range $(R)$, the sample standard deviation $(S)$, and the sample variance $\left(S^{2}\right)$.

Typically, we use a control chart to monitor the process mean together with a control chart to monitor the process variation. If the variation is IC, we go ahead and examine the control chart for the mean. For example, a Shewhart $\bar{X}$ chart for the mean is often used together with a Shewhart $R$ chart for the spread. Note that, for illustration, we consider the Shewhart $R$ chart even though recent literature recommends using a different spread chart, such as the Shewhart $S$ chart; see, for instance, Mahmoud et al. (2010). We do this because the Shewhart $R$ chart is simple and continues to be used in the industry.

In Case $\mathrm{K}$, the values of $\mu$ and $\sigma$ are known or are specified so that they can be used to construct the respective control charts. We illustrate the Shewhart $R$ and $S$ charts for the known standard deviation $\sigma_{0}$.

## 统计代写|统计推断代写STATISTICAL INFERENCE代写|CUSUM Control Charts

CUSUM 图表统计， $C_{i}^{+}$和 $C_{i}^{-}$，分别在表 $3.6$ 的 (b) 和 (d) 列中给出，而相应的量， $N^{+}$和 $N^{-}$分别在 (c) 和 (e) 栏中给出。为 了说明计算，考虑第 1 期。第一期的图表统计计算如下
$$C_{1}^{+}=\max \left[0, X_{1}-\left(\mu_{0}+K\right)+C_{0}^{+}\right]=\max [0,9.45-(10+0.5)+0] \quad=\max [0,-1.05]=0$$

$$C_{1}^{-}=\min \left[0, X_{1}-\mu_{0}+K+C_{0}^{-}\right]=\max [0,9.45-10+0.5+0] \quad=\max [0,-0.05]=0 .$$

## MATLAB代写

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