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# 统计代写|统计推断代写STATISTICAL INFERENCE代写|STS232 PARAMETRIC VARIABLES CONTROL CHARTS IN CASE K

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

Among the many control charts used in practice, the Shewhart charts are the most popular because of their simplicity, ease of application, and the fact that these versatile charts are quite efficient in detecting moderate to large shifts. These charts were originally proposed by Walter Shewhart in 1926. To describe the Shewhart chart in general, suppose that a process location parameter $\theta$, such as the mean, is to be monitored using a charting statistic $T$, which is a good point estimator of $\theta$, statistically speaking. Further suppose that the expected value and the standard deviation of $T$ are $\mu_{T}$ and $\sigma_{T}$, respectively. Statistical considerations often lead us to take $T$ to be an unbiased estimator of $\theta$ so that $\mu_{T}=\theta$. Then, a general formula for the center line $(C L)$ and the control limits of a Shewhart control chart are
\begin{aligned} U C L &=\theta+k \sigma_{T} \ C L &=\theta \ L C L &=\theta-k \sigma_{T} \end{aligned}
where $k>0$ is the charting constant, which is a chart design parameter that determines the “distance” of the control limits from the CL, expressed in terms of the standard deviation. Hence, these control limits are often called $k$-sigma limits. A Shewhart $k$-sigma control chart is the graphic that displays these three limits as straight lines along with the realized (calculated) values of the charting statistic $T$ for a number of samples or over time. Note that, in a Shewhart chart, the upper and the lower control limits are symmetrically placed around the $C L$. Such control limits are more meaningful when the distribution of $T$ is symmetric or approximately so, which goes well with the assumption in a Shewhart chart that either the process distribution is normal or that $T$ has a distribution that is approximately normal with mean $\theta$. Suppose, for example, that $\theta$ is the process mean $\mu$ to be monitored and the IC value of $\mu$ is $\mu_{0}$. In this case, $T$ is taken to be the mean $\bar{X}$ of the sample and then the $k$-sigma limits are given by $\mu_{0} \pm k \sigma_{0} / \sqrt{n}$, where $\sigma_{0}$ is the known process standard deviation, since $\sigma_{\bar{X}}=\sigma_{0} / \sqrt{n}$. The rationale behind the $k$-sigma limits is that $\bar{X}$ is exactly (or approximately) normally distributed when the process distribution is normal (or by virtue of the central limit theorem). When the charting statistic plots on or outside of either the upper or the lower control limits, we say that a signal has been observed or that a signaling event has taken place and the process is declared to be out-of-control (OOC).

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

While the Shewhart charts are widely known and often used in practice because of their simplicity and effectiveness in detecting moderate to large shifts, other charts, such as CUSUM charts, may be more useful in certain situations for detecting smaller, persistent kind of shifts. These charts, sometimes labeled time-weighted charts, are more naturally appropriate in the process control environment in view of the sequential nature of data collection. The CUSUM control charts were first introduced by Page (1954) (although not in its present form) and have been studied by many authors over the last 60 years. See, for example, Barnard (1959), Ewan and Kemp (1960), Johnson (1961), Goldsmith and Whitfield (1961), Page (1961), Ewan (1963), Hawkins (1992, 1993), and Hawkins and Olwell (1998). These are some examples, and since the introduction of CUSUM charts in 1954 by Page, there has been an incredible amount of work on CUSUM charts (see the overview in the Encyclopaedia of Statistics in Quality and Reliability by Ruggeri, Kenett, and Faltin (2007a) and the citations therein, for example). The CUSUM charts are typically based on the CUSUMs of a statistic or of differences of a statistic from its IC expected value, and are calculated progressively as the data accumulate over time. For example, the CUSUM chart for the mean is typically based on the CUSUM of the deviations of the individual observations (or the subgroup means) from the specified value of the IC target mean.

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

$$U C L=\theta+k \sigma_{T} C L \quad=\theta L C L=\theta-k \sigma_{T}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。