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# 统计代写|非参数统计代写Nonparametric Statistics代考|BIOSTAT885 Using the Estimator Sp

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## 统计代写|非参数统计代写Nonparametric Statistics代考|Using the Estimator Sp

Next, we illustrate the probability limits based Shewhart $S$ chart with $S_{p}$ as the estimator of $\sigma$. Note that, for this data set, the Phase I estimator $S_{p}=$ $\sqrt{\sum_{i=1}^{m} S_{i}^{2} / m}=0.1391$. From Table $\mathrm{H}$ in Appendix A, the needed charting constants are found to be $H_{3}^{}(25,5)=0.1581$ and $H_{4}^{}(25,5)=2.1240$ for a nominal $A R L_{\mathrm{IC}}=370$. Thus, the probability limits-based Shewhart $S$ chart is given by $L C L=H_{3}^{}(m, n) S_{p}=(0.1581)(0.1391)=0.0220$ and $U C L=H_{4}^{}(m, n) S_{p}=(2.1240)(0.1391)=0.2955$ with $C L=0.1391$. Note that this chart is slightly narrower than the probability limits-based Shewhart $S$ chart using the estimator $\bar{S} / c_{4}$ shown in Figure 3.26. This is expected since $S_{p}$ is a more efficient estimator than $\bar{S} / c_{4}$.

In Figure 3.27, all of the points plot between the control limits and there are no anomalous patterns in the data. Hence, we conclude that the process is functioning in statistical control with respect to variation.

## 统计代写|非参数统计代写Nonparametric Statistics代考|The Shewhart Chart for the Mean in Case U

The conditioning-unconditioning (CUC) method, which was first explicitly coined and used in Chakraborti (2000), is explained here for the parametric Shewhart $\bar{X}$ control chart for the mean, assuming the normal distribution. This development is important since the “standards unknown” case, that is, Case $\mathrm{U}$, is the situation often encountered in practice, and the Shewhart $\bar{X}$ control chart for the mean is one of the most popular charts used in practice. A brief background is given before going into detail. Recall that in the “standards known” case, that is, when process parameters are known (or Case $\mathrm{K}$ ), the signaling events are mutually independent so that the run-length distribution is geometric with the probability of a success (which, in $\mathrm{SPC}$, is a signal) equal to, say, some $\theta$. This result completely characterizes the performance of the Shewhart control chart in Case $K$, so that all performance properties of the chart can be obtained from the properties of the $G E O(\theta)$ distribution. Thus, the expected value of the run-length distribution is the reciprocal of $\theta$. For the IC run-length distribution, the $A R L_{\mathrm{IC}}$, simply equals the reciprocal of the $F A R$, which is the probability of a signal $\theta$ when the process is IC. The IC signal probability is denoted by $\alpha$ and therefore, $A R L_{\mathrm{TC}}=1 / \alpha$ so that specifying the $F A R$ specifies the $A R L_{\mathrm{TC}}$ and vice versa. This simple relationship makes understanding the performance of the Shewhart $\bar{X}$ chart easier in Case $\mathrm{K}$. Conversely, when the process parameters are unknown and need to be estimated to set up the control limits before Phase II process monitoring can begin, the signaling events are no longer independent so that the run-length distribution is no longer geometric. As a consequence, for example, the $A R L$ is no longer the reciprocal of the probability of a signal. This is the fundamental conceptual difference between Case $\mathrm{K}$ and Case U, which has important implications since some may find it tempting to use the results of Case $\mathrm{K}$ to design control charts even when the underlying process parameters are unknown. Practitioners are cautioned against this (see, for example, Quesenberry, 1993) practice and we thus discuss how to handle the situation properly. To this end, we use an important tool called the “conditioning-unconditioning method” proposed in Chakraborti (2000) and used extensively in the literature by many researchers. Note that Chen (1997), among others, has also used similar ideas.

## 统计代写|非参数统计代写NONPARAMETRIC STATISTICS代 考|USING THE ESTIMATOR SP

$L C L=H_{3}(m, n) S_{p}=(0.1581)(0.1391)=0.0220$ 和 $U C L=H_{4}(m, n) S_{p}=(2.1240)(0.1391)=0.2955$ 和 $C L=0.1391$. 请注意，此图表比基于概率限生

## Matlab代写

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