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# 统计代写|非参数统计代写Nonparametric Statistics代考|EDRS741 The Shewhart Chart for the Variance in Case U

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

The run-length distribution of the Shewhart charts for the variance can be obtained similarly, as in the case of the mean, using the CUC method. We sketch a brief example here for the $S$ chart with $S_{p}$ as the estimator of $\sigma$ and derive the IC run-length distribution.

Note that, in this case, the probability limits of the Phase II chart are given by $L C L=H_{3}^{}(m, n) S_{p}, C L=S_{p}$, and $U C L=H_{4}^{}(m, n) S_{p}$, where $H_{3}^{}(m, n)=\sqrt{\chi_{\alpha(m, n) / 2, n-1}^{2} /(n-1)}, H_{4}^{}=\sqrt{\chi_{1-\alpha(m, n) / 2, n-1}^{2} /(n-1)}$, and $\alpha(m, n)$ is the unconditional probability that the Phase II charting statistic, the standard deviation, $S_{i}$ plots outside either the lower or the upper control limit when the process is IC. Note that, in the parameter known case, this is the probability of a false alarm or the $F A R$ and it only depends on the sample size $n$, but in the unknown parameter case, the probability $\alpha(m, n)$ depends on both $m$ and $n$. Once the quantity $\alpha(m, n)$ is determined, the probability limits are easily obtained using the corresponding percentiles of the chi-square distribution with $n-1$ degrees of freedom. Although we may refer to $\alpha(m, n)$ as the FAR (probability of a signal when the process is $\mathrm{IC}$ ) in Case U, for convenience, note that it does not have the same clear interpretation as in Case $\mathrm{K}$ since the signaling events are dependent in Case U. Some details about the derivation of $\alpha(m, n)$ are given in Appendix 3.3. The $\alpha(m, n)$ values are calculated and shown in Table $\mathrm{H}$ in Appendix A along with the corresponding control limits $H_{3}^{}(m, n)$ and $H_{4}^{}(m, n)$ for $n=5$ and 10 and some values of $m$ from 5 to 100 , with nominal $A R L_{\mathrm{IC}}$ values of 370 and 500 . Note that the $\alpha(m, n)$ values depend both on $m$ and $n$, and for a given value of $n$, they increase for increasing values of $m$ (increasing the number of Phase I samples).
Other moments of the conditional and the unconditional IC run-length distribution can be found in a similar way. Note that when other estimators of $\sigma$, such as the one based on the $\bar{R}$ or the $\bar{S}$ are used in the Phase I control limits, or a different chart such as the $R$ chart is used in Phase II, a similar approach can be used to derive the run-length distribution and its various moments by applying the CUC method. We basically need the probability distributions of the charting statistic and of the Phase I estimator of $\sigma$. Note that the same approach can be also used to derive the OOC run-length distribution, which is useful in studying the performance of these charts in shift detection and comparisons. We leave the details to the reader.

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

The reader is referred back to Example $3.6$ in Section 3.6.1.2. In this example, the underlying process distribution is assumed to be $N(0,1)$. Let us consider, for example, the one-step transition probability, $p_{00}$. It was shown that $p_{00}=P\left(X_{t} \leq 1\right)=0.841$. However, recall that the $P\left(X_{t} \leq 1\right)=0.841$ was obtained using the cdf of a $N(0,1)$ distribution, which requires not only the knowledge of the normality of the process distribution but also that the mean and the standard deviation, assumed to be 0 and 1 , respectively. Suppose now that we are in Case KU, that is, the process mean is known (say, is equal to zero), but the process variance is unknown. In this case, we have a single unknown parameter $\sigma$ to be estimated by some estimator $T$ (which is typically obtained from a Phase I analysis) and we calculate the elements of the essential probability matrix $Q$, that is, the transition probabilities, conditionally, given the estimator $T$. Now $p_{00}=P\left(X_{t} \leq 1\right)$ is calculated as $\hat{p}{00}=P\left(X{t} \leq 1 \mid T\right)=\Phi(1 / T)$ where $T$ is some estimator of $\sigma$. The same applies to the other one-step transition probabilities. Thus, $Q(T)$, the estimated conditional essential probability matrix, is obtained and substituted into Equations 3.36-3.39. Finally, using the CUC method, the unconditional run-length distribution is obtained by taking expectation over the distribution of $T$, as explained above and illustrated in Equation $3.40$ for the unconditional ARL. Case UK follows similarly. However, this time the process mean needs to be estimated and dealt with and, for Case UU, where both the process mean and the process variance need to be estimated, we condition on, say, $T_{1}$ and $T_{2}$, where $T_{1}$ denotes the estimator for the process mean and $T_{2}$ denotes the estimator for the process variance. The reader is also referred to Jones, Champ, and Rigdon (2004) for an alternative derivation of the run-length distribution of the CUSUM chart with estimated parameters.

## 统计代写|非参数统计代写 NONPARAMETRIC STATISTICS代 考|THE SHEWHART CHART FOR THE VARIANCE IN CASE U

$H_{3}(m, n)=\sqrt{\chi_{\alpha(m, n) / 2, n-1}^{2} /(n-1)}, H_{4}=\sqrt{\chi_{1-\alpha(m, n) / 2, n-1}^{2} /(n-1)}$ ，和 $\alpha(m, n)$ 是第二阶段图表统计的无条件概率，标准偏差， $S_{i}$ 当过程为 IC 时，绘制在

370 和 500 的值。请注意， $\alpha(m, n)$ 值取决于两者 $m$ 和 $n$ ，并且对于给定的值 $n$, 官们随存值的增加而墁加 $m$ increasingthenumberof PhaseIsamples。 可以以类似的方式找到有条件和无条件 IC 游程分布的其他矩。请注意，当其他估计量 $\sigma$ ，例如基于 $\bar{R}$ 或者 $\bar{S}$ 用于第一阶段控制限制或不同的图表，例如 $R$ 图表用于

## 统计代写非参数统计代写NONPARAMETRIC STATISTICS代 考|THE CUSUM CHART FOR THE MEAN IN CASE U

whichistypicallyobtained fromaPhaseIanalysis㧴们计算㫷本概率矩阵的元责 $Q$ ，也就是说，转移概率，有条件地，给定估计量 $T$. 现在 $p_{00}=P\left(X_{t} \leq 1\right)$ 计

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