## 统计代写|回归分析作业代写Regression Analysis代考|STA4210

statistics-lab™ 为您的留学生涯保驾护航 在代写回归分析Regression Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写回归分析Regression Analysis代写方面经验极为丰富，各种代写回归分析Regression Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|回归分析作业代写Regression Analysis代考|Evaluating the Constant Variance

The first graph you should use to evaluate the constant variance assumption is the $\left(\hat{y}_i, e_i\right)$ scatterplot. Look for changes in the pattern of vertical variability of the $e_i$ for different $\hat{y}_i$. The most common indications of constant variance assumption violation are shapes that indicate either increasing variability of $Y$ for larger $\mathrm{E}(Y \mid X=x)$, or shapes that indicate decreasing variability of $Y$ for larger $\mathrm{E}(Y \mid X=x)$. Increasing variability of $Y$ for larger $\mathrm{E}(Y \mid X=x)$ is indicated by greater variability in the vertical ranges of the $e_i$ when $\hat{y}_i$ is larger.
Recall again that the constant variance assumption (like all assumptions) refers to the data-generating process, not the data. The statement “the data are homoscedastic” makes no sense. By the same logic, the statements “the data are linear” and “the data are normally distributed” also are nonsense. Thus, whichever pattern of variability that you decide to claim based on the $\left(\hat{y}_i, e_i\right)$ scatterplot, you should try to make sense of it in the context of the subject matter that determines the data-generating process. As one example, physical boundaries on data force smaller variance when the data are closer to the boundary. As another, when income increases, people have more choice as to whether or not they choose to purchase an item. Thus, there should be more variability in expenditures among people with more money than among people with less money. Whatever pattern you see in the $\left(\hat{y}_i, e_i\right)$ scatterplot should make sense to you from a subject matter standpoint.

While the LOESS smooth to the $\left(\hat{y}_i, e_i\right)$ scatterplot is useful for checking the linearity assumption, it is not useful for checking the constant variance assumption. Instead, you should use the LOESS smooth over the plot of $\left(\hat{y}_i,\left|e_i\right|\right)$. When the variability in the residuals is larger, they will tend to be farther from zero, giving larger mean absolute residuals $\left|e_i\right|$. An increasing trend in the $\left(\hat{y}_i,\left|e_i\right|\right)$ plot suggests larger variability in $Y$ for larger $\mathrm{E}(Y \mid X=x)$, and a flat trend line for the $\left(\hat{y}_i,\left|e_i\right|\right)$ plot suggests that the variability in $Y$ is nearly unrelated to $\mathrm{E}(Y \mid X=x)$. However, as always, do not over-interpret. Data are idiosyncratic (random), so even if homoscedasticity is true in reality, the LOESS fit to the $\left(\hat{y}_i,\left|e_i\right|\right)$ graph will not be a perfectly flat line, due to chance alone. To understand “chance alone” in this case you can simulate data from a homoscedastic model, construct the $\left(\hat{y}_i,\left|e_i\right|\right)$ graph, and add the LOESS smooth. You will see that the LOESS smooth is not a perfect flat line, and you will know that such deviations are explained by chance alone.

The hypothesis test for homoscedasticity will help you to decide whether the observed deviation from a flat line is explainable by chance alone, but recall that the test does not answer the real question of interest, which is “Is the heteroscedasticity so bad that we cannot use the homoscedastic model?” (That question is best answered by simulating data sets having the type of heteroscedasticity you expect with your real data, then by performing the types of analyses you plan to perform on your real data, then by evaluating the performance of those analyses.)

## 统计代写|回归分析作业代写Regression Analysis代考|Evaluating the Constant Variance Assumption

Consider the $\left(\hat{y}_i, \mid e_i\right)$ scatterplot in the right-hand panel of Figure 4.7. In that plot, there is an increasing trend that suggests heteroscedasticity. You can test for trend in the $\left(\hat{y}_i,\left|e_i\right|\right)$ scatterplot by fitting an ordinary regression line to those data, and then testing for significance of the slope coefficient. Significance $(p<0.05)$ means that the observed trend is not easily explained by chance alone under the homoscedastic model; insignificance $(p>0.05)$ means that the observed trend is explainable by chance alone under the homoscedastic model. This test is called the Glejser test (Glejser 1969).

There are many tests for heteroscedasticity other than the Glejser test, including the “Breusch-Pagan test” and “White’s test.” These tests use absolute and/or squared values of the residuals. Because absolute and squared residuals are non-negative, the assumption of normality of the absolute and squared residuals is obviously violated. Hence these tests are only approximately valid.

Another approach to testing heteroscedasticity is to model the variance function $\operatorname{Var}(Y \mid X=x)=g(x, \theta)$ explicitly within a model that uses a reasonable (perhaps nonnormal) distribution for $Y \mid X=x$, then to estimate the model using maximum likelihood, and then to test for constant variance in the context of that model using the likelihood ratio test. This approach is better because it identifies the nature of the heteroscedasticity explicitly, which may be an end unto itself in your research. This approach is also better because you can use the resulting heteroscedastic variance function $g(x, \theta)$ to obtain weighted least-squares (WLS) estimates of the $\beta$ ‘s that are better than the ordinary least-squares (OLS) estimates. Chapter 12 discusses these issues further.

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|回归分析作业代写Regression Analysis代考|STAT2220

statistics-lab™ 为您的留学生涯保驾护航 在代写回归分析Regression Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写回归分析Regression Analysis代写方面经验极为丰富，各种代写回归分析Regression Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|回归分析作业代写Regression Analysis代考|Evaluating the Linearity Assumption Using Graphical Methods

While we are not big fans of data analysis “recipes,” in regression or elsewhere, which instruct you to perform step 1, step 2, step 3, etc. for the analysis of your data, we are happy to recommend the following first step for the analysis of regression data.
Step 1 of any analysis of regression data
Plot the ordinary $\left(x_i, y_i\right)$ scatterplot, or scatterplots if there are multiple $X$ variables.
The simple $\left(x_i, y_i\right)$ scatterplot gives you immediate insight into the viability of the linearity, constant variance, and normality assumptions (see Section $1.8$ for examples of such scatterplots). It will also alert you to the presence of outliers.

To evaluate linearity using the $\left(x_i, y_i\right)$ scatterplot, simply look for evidence of curvature. You can overlay the LOESS fit to better estimate the form of the curvature. Recall, though, that all assumptions refer to the data-generating process. Thus, if you are going to claim there is curvature, such curvature should make sense in the context of the subject matter. For one example, boundary constraints can force curvature: If the minimum $Y$ is zero, then the curve must flatten for $X$ values where $Y$ is close to zero. For another example, in the case of the product preference vs. product complexity shown in Figure 1.16, there is a subject matter rationale for the curvature: People prefer more complexity up to a point, after which more complexity is less desirable. Ideally, you should be able to justify curvature in terms of the processes that produced your data.

A refinement of the $\left(x_i, y_i\right)$ scatterplot is the residual $\left(x_i, e_i\right)$ scatterplot. This scatterplot is an alternative, “magnified” view of the $\left(x_i, y_i\right)$ scatterplot, where the $e=0$ horizontal line in the $\left(x_i, e_i\right)$ scatterplot corresponds to the least-squares line in the $\left(x_i, y_i\right)$ scatterplot. Look for upward or downward ” $\mathrm{U}^{\prime \prime}$ shape to suggest curvature; overlay the LOESS fit to the $\left(x_i, e_i\right)$ data to help see these patterns.

You can also use the $\left(\hat{y}i, e_i\right)$ scatterplot to check the linearity assumption. In simple regression (i.e., one $X$ variable), the $\left(\hat{y}_i, e_i\right)$ scatterplot is identical to the $\left(x_i, e_i\right)$ scatterplot, with the exception that the horizontal scale is linearly transformed via $\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1 x_i$. When the estimated slope is negative, the horizontal axis is “reflected”-large values of $x$ map to small values of $\hat{y}_i$ and vice versa. You can use this plot just like the $\left(x_i, e_i\right)$ scatterplot. In simple regression, the $\left(\hat{y}_i, e_i\right)$ scatterplot offers no advantage over the $\left(x_i, e_i\right)$ scatterplot. However, in multiple regression, the $\left(\hat{y}_i, e_i\right)$ scatterplot is invaluable as a quick look at the overall model, since there is just one $\left(\hat{y}_i, e_i\right)$ plot to look at, instead of several $\left(x{i j}, e_i\right)$ plots (one for each $X_j$ variable). This $\left(\hat{y}_i, e_i\right)$ scatterplot, which you can call a “predicted/residual scatterplot,” is automatically provided by $\mathrm{R}$ when you plot a fitted lm object.

## 统计代写|回归分析作业代写Regression Analysis代考|Evaluating the Linearity Assumption Using Hypothesis Testing Methods

Here, we will get slightly ahead of the flow of the book, because multiple regression is covered in the next chapter. A simple, powerful way to test for curvature is to use a multiple regression model that includes a quadratic term. The quadratic regression model is given by:
$$Y=\beta_0+\beta_1 X+\beta_2 X^2+\varepsilon$$
This model assumes that, if there is curvature, then it takes a quadratic form. Logic for making this assumption is given by “Taylor’s Theorem,” which states that many types of curved functions are well approximated by quadratic functions.

Testing methods require restricted (null) and unrestricted (alternative) models. Here, the null model enforces the restriction that $\beta_2=0$; thus the null model states that the mean response is a linear (not curved) function of $x$. So-called “insignificance” (determined historically by $p>0.05$ ) of the estimate of $\beta_2$ means that the evidence of curvature in the observed data, as indicated by a non-zero estimate of $\beta_2$ or by a curved LOESS fit, is explainable by chance alone under the linear model. “Significance” (determined historically by $p<0.05$ ) means that such evidence of curvature is not easily explained by chance alone under the linear model.

But you should not take the result of this $p$-value based test as a “recipe” for model construction. If “significant,” you should not automatically assume a curved model. Instead, you should ask, “Is the curvature dramatic enough to warrant the additional modeling complexity?” and “Do the predictions differ much, whether you use a model for curvature or the ordinary linear model?” If the answers to those questions are “No,” then you should use the linear model anyway, even if it was “rejected” by the $p$-value based test.

In addition, models employing curvature (particularly quadratics) are notoriously poor at the extremes of the $x$-range(s). So again, you can easily prefer the linear model, even if the curvature is “significant” $(p<0.05)$.

Conversely, if the quadratic term is “insignificant,” it does not mean that the function is linear. Recall from Chapter 1 that the linearity is usually false, a priori; hence, “insignificance” means that you have failed to detect curvature. If the test for the quadratic term is “insignificant,” it is most likely a Type II error.

Even when the curvature does not have a perfectly quadratic form, the quadratic test is usually very powerful; rare exceptions include cases where the curvature is somewhat exotic. If the quadratic model is grossly wrong for modeling curvature in your application, then you should use a test based on a model other than the quadratic model.

## 统计代写|回归分析作业代写Regression Analysis代考|Evaluating the Linearity Assumption Using Hypothesis Testing Methods

$$Y=\beta_0+\beta_1 X+\beta_2 X^2+\varepsilon$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|回归分析作业代写Regression Analysis代考|STA321

statistics-lab™ 为您的留学生涯保驾护航 在代写回归分析Regression Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写回归分析Regression Analysis代写方面经验极为丰富，各种代写回归分析Regression Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|回归分析作业代写Regression Analysis代考|Descriptive Methods Versus Testing Methods for Checking Assumptions

One benefit of using graphical/descriptive methods to check assumptions, rather than hypothesis testing ( $p$-value based) methods, is transparency: The graphs show the data, as they are. The $p$-values of the statistical tests give information that is distorted by the sample size. Another benefit is that you can determine the practical significance of a result using graphical methods and descriptive statistics, but not by statistical tests and their $p$-values. Tests can tell you whether a result is statistically significant (again, historically, $p<0.05$ ), but statistically significant results can be practically unimportant, and vice versa, because of the sample size distortion. Unlike statistical tests of assumptions, larger sample sizes always point you closer to the best answer when you use well-chosen graphs and descriptive statistics.

But, care is needed in interpreting and constructing graphs. Interpreting graphs requires practice, judgment, and some knowledge of statistics. In addition, producing good graphs requires skill, practice, and in some cases, an artistic eye. A classic and very helpful text on the use and construction of statistical graphics is The Visual Display of Quantitative Information, by Edward Tufte (Tufte 2001).

The only good thing about tests is that they answer the question, “Is the apparent deviation from the assumption that is seen in the data explainable by chance alone?” The question of whether a result is explainable by chance alone is indeed important because researchers are prone to over-interpret idiosyncratic (chance) aspects of their data. Hypothesis testing provides a reality check to guard against such over-interpretation. But other methods, simulation in particular, are better for assessing the effects of chance deviation. Hence, $p$-value based hypothesis testing methods are not even needed for their one use, which is to assess the effect of chance variation.

Tests of model assumptions have been used for much of statistical history and are still used today in some quarters. Perhaps the main reason for their historical persistence is simplicity. Researchers have routinely applied the rule, “p-value greater than $0.05 \rightarrow$ assumption is satisfied; $p$-value less than $0.05 \rightarrow$ assumption is not satisfied,” because it is simple, despite it being a horribly misguided practice. We have already mentioned many concerns with tests, but here they are, in set-off form, so that you can easily refer to them.

## 统计代写|回归分析作业代写Regression Analysis代考|Which Assumptions Should You Evaluate First

We suggest (only mildly; this is not a hard-and-fast rule) that you evaluate the linearity and constant variance assumptions first. The reason is that, for checking the assumptions of independence and normality, you often will use the residuals $e_i=y_i-\hat{y}_{i,}$, where the predicted values $\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1 x_i$ are based on the linear fit. If the assumption of linearity is badly violated, then these estimated residuals will be badly biased. In such a case you should evaluate the normality and independence assumptions by first fitting a more appropriate (non-linear) model, and then by using that model to calculate the predicted values and associated residuals.

Furthermore, if the linearity assumption is reasonably valid but the homoscedasticity (constant variance) assumption is violated, then the residuals $e_i$ will automatically look non-normal, even when the conditional distributions $p(y \mid x)$ are normal because some residuals will come from distributions with larger variance and some will come from distributions with smaller variance, lending a heavy-tailed appearance to the pooled $\left{e_i\right}$ data. For these reasons, we mildly suggest that you evaluate the assumptions in the order (1) linearity, (2) constant variance, (3) independence, and (4) normality. But there are cases where this sequence is logically flawed, so please just treat it as one of those “ugly rules of thumb.”

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|抽样调查作业代写sampling theory of survey代考|STAT506

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|抽样调查作业代写sampling theory of survey代考|EXAMPLES OF REPRESENTATIVE STRATEGIES

The ratio estimator
$$t_{1}=X \frac{\sum_{i \in s} Y_{i}}{\sum_{i \in s} X_{i}}$$
is of special importance because of its traditional use in practice. Here, $\left(p, t_{1}\right)$ is obviously representative with respect to a size measure $x$, more precisely to $\left(X_{1}, \ldots, X_{N}\right)$, whatever the sampling design $p$.

Note, however, that $t_{1}$ is usually combined with SRSWOR or SRSWR. The sampling scheme of LAHIRI-MIDZUNO-SEN (LAHIRI, 1951; MIDZUNO, 1952; SEN, 1953) (LMS) yields a design of interest to be employed in conjunction with $t_{1}$ by rendering it design unbiased.
The Hansen-Hurwitz (HH, 1943) estimator (HHE)
$$t_{2}=\frac{1}{n} \sum_{i=1}^{N} f_{s i} \frac{Y_{i}}{P_{i}}$$ with $f_{s i}$ as the frequency of $i$ in $s, i \in \mathcal{U}$, combined with any design $p$, gives rise to a strategy representative with respect to $\left(P_{1}, \ldots, P_{N}\right)^{\prime}$. For the sake of design unbiasedness, $t_{2}$ is usually based on probability proportional to size (PPS) with replacement (PPSWR) sampling, that is, a scheme that consists of $n$ independent draws, each draw selecting unit $i$ with probability $P_{i}$.

Another representative strategy is due to RAO, HARTLEY and COCHRAN (RHC, 1962). We first describe the sampling scheme as follows: On choosing a sample size $n$, the population $\mathcal{U}$ is split at random into $n$ mutually exclusive groups of sizes suitably chosen $N_{i}\left(i=1, \ldots, n ; \sum_{1}^{n} N_{i}=N\right)$ coextensive with $\mathcal{U}$, the units bearing values $P_{i}$, the normed sizes $\left(0<P_{i}<1, \sum P_{i}=1\right)$. From each of the $n$ groups so formed independently one unit is selected with a probability proportional to its size given the units falling in the respective groups.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Raj’s Estimator t5

Another popular strategy is due to RAJ $(1956,1968)$. The sampling scheme is called probability proportional to size without replacement (PPSWOR) with $P_{i}$ ‘s $\left(02)$ draw a unit $i_{n}\left(\neq i_{1}, \ldots, i_{n-1}\right)$ is chosen with probability
$$\frac{P_{i_{n}}}{1-P_{i_{1}}-P_{i_{2}}-\ldots,-P_{i_{n-1}}}$$ out of the units of $U$ minus $i_{1}, i_{2}, \ldots, i_{n-1}$. Then,
\begin{aligned} e_{1} &=\frac{Y_{i_{1}}}{P_{i_{1}}} \ e_{2} &=Y_{i_{1}}+\frac{Y_{i_{2}}}{P_{i_{2}}}\left(1-P_{i_{1}}\right) \ e_{j} &=Y_{i_{1}}+\ldots+Y_{i_{j-1}}+\frac{Y_{i_{j}}}{P_{i_{j}}}\left(1-P_{i_{1}}-\ldots-P_{i_{j-1}}\right) \end{aligned}
$j=3, \ldots, n$ are all unbiased for $Y$ because the conditional expectation
\begin{aligned} E_{c} & {\left[e_{j} \mid\left(i_{1}, Y_{i_{1}}\right), \ldots,\left(i_{j-1}, Y_{i_{j-1}}\right)\right] } \ &=\left(Y_{i_{1}}+\ldots,+Y_{i_{j-1}}\right)+\sum_{\substack{k=1 \ \left(\neq i_{1}, \ldots, i_{j-1}\right)}}^{N} Y_{k}=Y . \end{aligned}
So, unconditionally, $E_{p}\left(e_{j}\right)=Y$ for every $j=1, \ldots, n$, and
$$t_{5}=\frac{1}{n} \sum_{j=1}^{n} e_{j},$$
called Raj’s (1956) estimator, is unbiased for $Y$.

## 统计代写|抽样调查作业代写sampling theory of survey代考|EXAMPLES OF REPRESENTATIVE STRATEGIES

$$t_{1}=X \frac{\sum_{i \in s} Y_{i}}{\sum_{i \in s} X_{i}}$$

$$t_{2}=\frac{1}{n} \sum_{i=1}^{N} f_{s i} \frac{Y_{i}}{P_{i}}$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Raj’s Estimator t5

$$\frac{P_{i_{n}}}{1-P_{i_{1}}-P_{i_{2}}-\ldots,-P_{i_{n-1}}}$$

$$e_{1}=\frac{Y_{i_{1}}}{P_{i_{1}}} e_{2}=Y_{i_{1}}+\frac{Y_{i_{2}}}{P_{i_{2}}}\left(1-P_{i_{1}}\right) e_{j}=Y_{i_{1}}+\ldots+Y_{i_{j-1}}+\frac{Y_{i_{j}}}{P_{i_{j}}}\left(1-P_{i_{1}}-\ldots-P_{i_{j-1}}\right)$$
$j=3, \ldots, n$ 都是公正的 $Y$ 因为条件期望
$$E_{c}\left[e_{j} \mid\left(i_{1}, Y_{i_{1}}\right), \ldots,\left(i_{j-1}, Y_{i_{j-1}}\right)\right]=\left(Y_{i_{1}}+\ldots,+Y_{i_{j-1}}\right)+\sum_{k=1} \sum_{\left(\neq i_{1}, \ldots, i_{j-1}\right)}^{N} Y_{k}=Y .$$

$$t_{5}=\frac{1}{n} \sum_{j=1}^{n} e_{j}$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|抽样调查作业代写sampling theory of survey代考|STAT7124

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|抽样调查作业代写sampling theory of survey代考|SAMPLING SCHEMES

A unified theory is developed by noting that it is enough to establish results concerning $(p, t)$ without heeding how one may actually succeed in choosing samples with preassigned probabilities. A method of choosing a sample draw by draw, assigning selection probabilities with each draw, is called a sampling scheme. Following HANURAV (1966), we show below that starting with an arbitrary design we may construct a sampling scheme.

Suppose for each possible sample $s$ from $U$ the selection probability $p(s)$ is fixed. Let
$$\begin{array}{lll} \beta_{i 1}=p\left(i_{1}\right), & \beta_{i_{1}, i_{2}}=p\left(i_{1}, i_{2}\right), \ldots, & \beta_{i_{1}, \ldots, i_{n}}=p\left(i_{1}, \ldots, i_{n}\right) \ \alpha_{i 1}=\Sigma_{1} p(s), & \alpha_{i_{1}, i_{2}}=\Sigma_{2} p(s), \ldots, & \alpha_{i_{1}, \ldots, i_{n}}=\Sigma_{n} p(s) \end{array}$$
where $\Sigma_{1}$ is the sum over all samples $s$ with $i_{1}$ as the first entry; $\Sigma_{2}$ is the sum over all samples with $i_{1}, i_{2}$, respectively, as the first and second entries in $s, \ldots$, and $\Sigma_{n}$ is the sum over all samples of which the first, second, $\ldots, n$th entries are, respectively, $i_{1}, i_{2}, \ldots, i_{n}$.

Then, let us consider the scheme of selection such that on the first draw from $U, i_{1}$ is chosen with probability $\alpha_{i 1}$, a second draw from $U$ is made with probability
$$\left(1-\frac{\beta_{i 1}}{\alpha_{i 1}}\right) \text {. }$$
On the second draw from $U$ the unit $i_{2}$ is chosen with probability
$$\begin{gathered} \alpha_{i_{1}, i_{2}} \ \alpha_{i 1}-\beta_{i 1} \end{gathered}$$
A third draw is made from $U$ with probability
$$\left(1-\frac{\beta_{i_{1}, i_{2}}}{\alpha_{i_{1}, i_{2}}}\right)$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|CONTROLLED SAMPLING

Now, consider an arbitrary design $p$ of fixed size $n$ and a linear estimator $t$; suppose a subset $S_{0}$ of all samples is less desirable from practical considerations like geographical location, inaccessibility, or, more generally, costliness. Then, it is advantageous to replace design $p$ by a modified one, for example, $q$, which attaches minimal values $q(s)$ to the samples $s$ in $S_{0}$ keeping
$$\begin{gathered} E_{p}(t)=E_{q}(t) \ E_{p}(t-Y)^{2}-E_{q}(t-Y)^{2} \end{gathered}$$
and even maintaining other desirable properties of $p$, if any. A resulting $q$ is called a controlled design and a corresponding scheme of selection is called a controlled sampling scheme. Quite a sizeable literature has grown around this problem of finding appropriate controlled designs. The methods of implementing such a scheme utilize theories of incomplete block designs and predominantly involve ingeneous devices of reducing the size of support of possible samples demanding trials and errors. But RAO and NIGAM (1990) have recently presented a simple solution by posing it as a linear programming problem and applying the well-known simplex algorithm to demonstrate their ability to work out suitable controlled schemes.
Taking $t$ as the HOR VIT7-THOMPSON estimator $\bar{t}=\sum_{i \in S}$ $Y_{i} / \pi_{i}$, they minimize the objective function $F=\sum_{s \in S_{0}} q(s)$ subject to the linear constraints
\begin{aligned} \sum_{s \ni i, j} q(s) &=\sum_{s \ni i, j} p(s)=\pi_{i j} \ q(s) & \geq 0 \text { for all } s \end{aligned}
where $\pi_{i j}{ }^{\prime} s$ are known quantities in terms of the original uncontrolled design $p$.

## 统计代写|抽样调查作业代写sampling theory of survey代考|SAMPLING SCHEMES

$$\beta_{i 1}=p\left(i_{1}\right), \quad \beta_{i_{1}, i_{2}}=p\left(i_{1}, i_{2}\right), \ldots, \quad \beta_{i_{1}, \ldots, i_{n}}=p\left(i_{1}, \ldots, i_{n}\right) \alpha_{i 1}=\Sigma_{1} p(s), \quad \alpha_{i_{1}, i_{2}}=\Sigma_{2} p(s), \ldots,$$

$$\left(1-\frac{\beta_{i 1}}{\alpha_{i 1}}\right) .$$

$$\alpha_{i_{1}, i_{2}} \alpha_{i 1}-\beta_{i 1}$$

$$\left(1-\frac{\beta_{i_{1}, i_{2}}}{\alpha_{i_{1}, i_{2}}}\right)$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|CONTROLLED SAMPLING

$$E_{p}(t)=E_{q}(t) E_{p}(t-Y)^{2}-E_{q}(t-Y)^{2}$$

$$\sum_{s \ni i, j} q(s)=\sum_{s \ni i, j} p(s)=\pi_{i j} q(s) \quad \geq 0 \text { for all } s$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|抽样调查作业代写sampling theory of survey代考|MATH525

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|抽样调查作业代写sampling theory of survey代考|ELEMENTARY DEFINITIONS

Let $N$ be a known number of units, e.g., godowns, hospitals, or income earners, each assignable identifying labels $1,2, \ldots, N$ and bearing values, respectively, $Y_{1}, Y_{2}, \ldots, Y_{N}$ of a realvalued variable $y$, which are initially unknown to an investigator who intends to estimate the total
$$Y=\sum_{1}^{N} Y_{i}$$
or the mean $\bar{Y}=Y / N$.
We call the sequence $U=(1, \ldots, N)$ of labels a population. Selecting units leads to a sequence $s=\left(i_{1}, \ldots, i_{n}\right)$, which is called a sample. Here $i_{1}, \ldots, i_{n}$ are elements of $U$, not necessarily distinct from one another but the order of its appearance is maintained. We refer to $n=n(s)$ as the size of $s$, while the effective sample size $v(s)=|s|$ is the cardinality of $s$, i.e., the number of distinct units in $s$. Once a specific sample $s$ is chosen we suppose it is possible to ascertain the values $Y_{i_{1}}, \ldots, Y_{i_{n}}$ of $y$ associated with the respective units of $s$. Then $d=\left[\left(i_{1}, Y_{i_{1}}\right), \ldots,\left(i_{n}, Y_{i_{n}}\right)\right] \quad$ or briefly $d=\left[\left(i, Y_{i}\right) \mid i \in s\right]$
constitutes the survey data.
An estimator $t$ is a real-valued function $t(d)$, which is free of $Y_{i}$ for $i \notin s$ but may involve $Y_{i}$ for $i \in s$. Sometimes we will express $t(d)$ alternatively by $t(s, Y)$, where $Y=\left(Y_{1}, \ldots\right.$, $\left.Y_{N}\right)^{\prime} .$

## 统计代写|抽样调查作业代写sampling theory of survey代考|DESIGN-BASED INFERENCE

Let $\Sigma_{1}$ be the sum over samples for which $|t(s, Y)-Y| \geq k>0$ and let $\Sigma_{2}$ be the sum over samples for which $|t(s, Y)-Y|<k$ for a fixed $Y$. Then from
\begin{aligned} M_{p}(t) &=\Sigma_{1} p(s)(t-Y)^{2}+\Sigma_{2} p(s)(t-Y)^{2} \ & \geq k^{2} \operatorname{Prob}[|t(s, Y)-Y| \geq k] \end{aligned}
one derives the Chebyshev inequality:
$$\operatorname{Prob}[|t(s, Y)-Y| \geq k] \leq \frac{M_{p}(t)}{k^{2}} .$$
Hence
$\operatorname{Prob}[t-k \leq Y \leq t+k] \geq 1-\frac{M_{p}(t)}{k^{2}}=1-\frac{1}{k^{2}}\left[V_{p}(t)+B_{p}^{2}(t)\right]$ where $B_{p}(t)=E_{p}(t)-Y$ is the bias of $t$. Writing $\sigma_{p}(t)=$ $\sqrt{V_{p}(t)}$ for the standard error of $t$ and taking $k=3 \sigma_{p}(t)$, it follows that, whatever $Y$ may be, the random interval $t \pm 3 \sigma_{p}(t)$ covers the unknown $Y$ with a probability not less than
$$\frac{8}{9}-\frac{1}{9} \frac{B_{p}^{2}(t)}{V_{p}(t)} .$$
So, to keep this probability high and the length of this covering interval small it is desirable that both $\left|B_{p}(t)\right|$ and $\sigma_{p}(t)$ be small, leading to a small $M_{p}(t)$ as well.

## 统计代写|抽样调查作业代写sampling theory of survey代考|ELEMENTARY DEFINITIONS

$$Y=\sum_{1}^{N} Y_{i}$$

Ibegin{aligned}
$\mathrm{M}{-}{\mathrm{p}}(\mathrm{t}) \&=\mid$ sigma ${1} \mathrm{p}(\mathrm{s})(\mathrm{t} \mathbf{\mathrm { Y }}) \wedge{2}+\backslash \operatorname{sigma}{2} \mathrm{p}(\mathrm{s})(\mathrm{tY}) \wedge{2} \backslash$
\& Igeq $k \wedge{2}$ loperatorname{概率 $}[|t(s, Y)-Y| \backslash g e q ~ k]$
lend{对齐 $}$
onederivestheChebyshevinequality:
loperatorname{概率 $}[|\mathrm{t}(\mathrm{s}, Y)-\mathrm{Y}| \operatorname{lgeq} \mathrm{k}] \backslash \operatorname{leq} \backslash f$ frac $\left{\mathrm{M}{-}{\mathrm{p}}(\mathrm{t})\right}{\mathrm{k} \wedge{2}}$ $\$ \$$因此 \operatorname{Prob}[t-k \leq Y \leq t+k] \geq 1-\frac{M{p}(t)}{k^{2}}=1-\frac{1}{k^{2}}\left[V_{p}(t)+B_{p}^{2}(t)\right] 在哪里 B_{p}(t)=E_{p}(t)-Y 是偏差 t. 写作 \sigma_{p}(t)=\sqrt{V_{p}(t)} 对于标准误 t 并采取 k=3 \sigma_{p}(t), 由此可知，无论 \ צmaybe, therandomintervalt Ipm 3 ปsigma_{p}(t)coverstheunknown 是withaprobabilitynotlessthan \frac{8}{9}-\frac{1}{9} \frac{B_{p}^{2}(t)}{V_{p}(t)}. So, tokeepthisprobabilityhighandthelengthofthiscoveringintervalsmallitisdesirablethatboth Veft \mid \mathrm{B}{-}{\mathrm{p}}(\mathrm{t}) \backslash right \mid and \backslash sigma{p}(t)besmall, leadingtoasmall \mathrm{M}_{-}{\mathrm{p}}(\mathrm{t}) \$$ 也是如此。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MA53200

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机分析作业代写stochastic analysis代写|Loynes’s scheme

Here we will consider the case where the state space $E$ is equipped with a partial ordering $\preceq$ (see section A.3), and admits a minimal point $\mathbf{0}$ such that $\mathbf{0} \preceq x$ for all $x \in E$. We will assume that on $E$ there exists a metric $d_{E}$ such that all $\preceq$-increasing sequences converge in $\bar{E}$, the adherence of $E$.
DEFINITION 2.5.- A function $\varphi: E \times F^{\mathbf{Z}} \rightarrow E$ is said $\preceq$-increasing when
$$\eta \preceq \eta^{\prime} \Longrightarrow \varphi(\eta, \omega) \preceq \varphi\left(\eta^{\prime}, \omega\right), \mathbf{P}{X}-a . s . .$$ It is said continuous with respect to its first variable when for $\mathbf{P}{X}$-almost all $\omega$, the function $(\eta \mapsto \varphi(\eta, \omega))$ is continuous for the metric $d_{E}$.

THEOREM $2.4$ (LOYNES’s THEOREM).- If $\varphi$ is $\preceq$-increasing and continuous, the equation [2.7] admits a solution $M_{\infty}$ with values in the adherence $\bar{E}$ of $E$.

Proof. Let us recall that we have assumed that we know the stimulus through the quadruple $\mathfrak{O}$, whose generic element is denoted $\omega$. We look for a random variable $Y$ valued in $E$ and satisfying [2.7]. We will get $Y$ as the limit of a sequence converging almost surely. To do this, we consider Loynes’s sequence $\left(M_{n}, n \in \mathbf{N}\right)$, defined by
$$M_{0}(\omega)=\mathbf{0} \text { and } M_{n+1}(\omega)=\varphi\left(M_{n} \circ \theta^{-1}(\omega), \theta^{-1} \omega\right), \forall n \geq 1 .$$
By the definition of $\mathbf{0}$, we have $M_{0}=\mathbf{0} \preceq M_{1}$, and assuming that for some $n>1$, $M_{n-1} \preceq M_{n}$ a.s., since $\varphi$ is increasing we have
$$M_{n}(\omega)=\varphi\left(M_{n-1}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right) \preceq \varphi\left(M_{n}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right)=M_{n+1}(\omega) \mathbf{P}_{X} \text {-a.s.. }$$

## 统计代写|随机分析作业代写stochastic analysis代写|Coupling

The idea of coupling plays a central role in the asymptotic study of SRS. It is in fact possible to state the conditions under which the trajectories of two SRS (or possibly those of the corresponding backward schemes) coincide at a certain point. These properties imply naturally, in particular, more traditional properties of convergence for random sequences such as convergence in distribution.

Hereafter we only state the results that will be useful to us in the applications to queueing, in their simplest form.

Secondly, we develop the theory of renovating events of Borovkov, which gives sufficient conditions for coupling, and even strong backward coupling. In addition, the results of Borovkov and Foss also allow in many cases to solve the equation [2.7], even when the conditions of continuity and monotonicity of Theorem $2.4$ are not satisfied. Particularly, in this framework we can also deal with the intricate question of the transient behavior depending on the initial conditions. In what follows, $\mathfrak{O}=$ $(\Omega, \mathcal{F}, \mathbf{P}, \theta)$ is a stationary ergodic quadruple.

## 统计代写|随机分析作业代写stochastic analysis代写|Loynes’s scheme

$$\eta \preceq \eta^{\prime} \Longrightarrow \varphi(\eta, \omega) \preceq \varphi\left(\eta^{\prime}, \omega\right), \mathbf{P} X-a . s . .$$

$$M_{0}(\omega)=\mathbf{0} \text { and } M_{n+1}(\omega)=\varphi\left(M_{n} \circ \theta^{-1}(\omega), \theta^{-1} \omega\right), \forall n \geq 1$$

$$M_{n}(\omega)=\varphi\left(M_{n-1}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right) \preceq \varphi\left(M_{n}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right)=M_{n+1}(\omega) \mathbf{P}_{X} \text {-a.s.. }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MATH477

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机分析作业代写stochastic analysis代写|Fluid model

A fluid model consists of replacing a queue which is a discrete-time event system by a reservoir of infinite capacity which empties itself at unit speed and is fed by some continuous data flow. We can then obtain qualitative results on models whose study supports no other approaches. On the one hand, the method does not require precise knowledge about the rate of the input process, and on the other hand, it is particularly well adapted to the study of extreme cases: low and high loads, superposition of heterogeneous traffic.

We work in continuous time and we assume that all the processes are rightcontinuous with left limits. We denote:
1) $S(t)$ : the total service time for the requests arrived up to time $t$;
2) $W(t)$ : the virtual waiting time of a customer arriving at time $t$, that is the time that the customer must wait before starting to be served;
3) $X(t)=S(t)-t$.
As the system has no losses, we have
$$W(t)=X(t)-\left(t-\int_{0}^{t} \mathbf{1}_{{0}}(W(s)) \mathrm{d} s\right) .$$
We will focus on showing an equivalent formulation of this equation.

## 统计代写|随机分析作业代写stochastic analysis代写|Canonical space

The concept of stationarity implies invariance in time, that is : a shift in time does not change the global picture. If the idea is easily understood, its formalization quickly clouds the basic concept.

Let us consider the set $F^{\mathbf{N}}$ of sequences of elements of a set $F$. The shift operator $\theta$ on $F^{\mathbf{N}}$ is then defined by
$$\theta: \begin{cases}F^{\mathbf{N}} & \longrightarrow F^{\mathbf{N}} \ \left(\omega_{n}, n \geq 0\right) & \longmapsto\left(\omega_{n+1}, n \geq 0\right)=\left(\omega_{n}, n \geq 1\right)\end{cases}$$
Defined in this way, this operator has the drawback of not being bijective: if we consider a sequence $\beta=\left(\beta_{n}, n \geq 0\right)$, all the sequences obtained by concatenation of any element of $F$ and $\beta$ are mapped onto $\beta$ by $\theta$. To overcome this problem, it is customary to work with sequences indexed by $\mathbf{Z}$ and not by $\mathbf{N}$. This change has no crucial mathematical consequence, as the indexation space remains countable. Philosophically, however, it implies that there is no more origin of time…
The shift operator is thus defined on $F^{\mathbf{Z}}$ by
$$\theta\left(\omega_{n}, n \in \mathbf{Z}\right)=\left(\omega_{n+1}, n \in \mathbf{Z}\right)$$
and thus becomes bijective!

## 统计代写|随机分析作业代写stochastic analysis代写|Fluid model

1) $S(t)$ ：请求的总服务时间到达时间 $t$;
2) $W(t)$ ：客户到达时间的虚拟等待时间 $t$ ，即客户在开始服务之前必须等待的时间；
3) $X(t)=S(t)-t$.

$$W(t)=X(t)-\left(t-\int_{0}^{t} \mathbf{1}_{0}(W(s)) \mathrm{d} s\right) .$$

## 统计代写|随机分析作业代写stochastic analysis代写|Canonical space

$$\theta:\left{F^{\mathbf{N}} \longrightarrow F^{\mathbf{N}}\left(\omega_{n}, n \geq 0\right) \longmapsto\left(\omega_{n+1}, n \geq 0\right)=\left(\omega_{n}, n \geq 1\right)\right.$$

$$\theta\left(\omega_{n}, n \in \mathbf{Z}\right)=\left(\omega_{n+1}, n \in \mathbf{Z}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|STAT342

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机分析作业代写stochastic analysis代写|Traffic, load, Erlang, etc.

In electricity, we count the amps or volts; in meteorology, we measure the pressure; in telecommunications, we count the Erlangs.

The telephone came into existence in 1870. Most of the concepts and notations were derived during this period. Looking at a telephone connection over a time period of length $T$, we define its observed traffic flow as the percentage of time during which the connection is busy
$$\rho=\frac{\sum_{i} t_{i}}{T}$$
A priori, traffic is a dimensionless quantity since it is the ratio of the occupation time to the total time. However, it still has a unit, Erlang, in remembrance of Erlang who, along with Palm, was one of the pioneers of the performance assessment of telephone networks. Therefore, a load of 1 Erlang corresponds to an always busy connection.

Looking at several connections, the traffic carried by this trunk is the sum of the traffic of each connection
$$\rho_{\text {trunk }}=\sum_{\text {connections }} \rho_{\text {connection }}$$
This is no longer a percentage, but we can give a physical interpretation to this quantity according to the ergodic hypothesis. In fact, assume that the number of junctions is large, then we can calculate the average occupation rate in two different ways: either by calculating the percentage of the occupation time of a particular connection over a large period of time; or by computing the percentage of busy connections at a given time.

## 统计代写|随机分析作业代写stochastic analysis代写|Lindley and Beneˇs

We often consider the number of customers present in the system but the quantity that contains the most information is the system load, defined at each moment as the time required for the system to empty itself in the absence of new arrivals. The server works at unit speed: it serves a unit of work per unit time. Consequently, the load decreases with speed 1 between two arrivals. Figure $1.8$ which represents the load over time depending on the arrivals and required service times is easily constructed.

DEFINITION 1.2.- A busy period of a queue is a period that begins with the arrival of a customer in an empty system (server plus buffer) and ends with the end of a service after which the system is empty again.

A cycle is a time period that begins with the arrival of a customer in an empty system and ends on the next arrival of a customer in an empty system. This is the concatenation of a busy period and an idle period, that is the time elapsed between the departure of the last customer of the busy period and the arrival of the next customer.

NOTE.- In Figure 1.8, a busy period begins at $T_{1}$ and ends at $D_{4}$. The corresponding cycle begins at $T_{1}$ and ends at $T_{5}$.

Note that as long as a service policy is conservative, the size of a busy period is independent of it: for waiting rooms of infinite size, the busy periods have, for example, the same length for the FIFO policy as that for the non-preemptive or preemptive resume LIFO policy.

## 统计代写|随机分析作业代写stochastic analysis代写|Traffic, load, Erlang, etc.

$$\rho=\frac{\sum_{i} t_{i}}{T}$$

$$\rho_{\text {trunk }}=\sum_{\text {connections }} \rho_{\text {connection }}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|运筹学作业代写operational research代考|KMA355

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|运筹学作业代写operational research代考|Normalform und kanonische Form

Um die Gleichungsrestriktionen von $P=$ in Matrix-Vektor-Form darstellen zu können, erweitern wir die Koeffizientenmatrix $A$ um die $(m, m)$-Einheitsmatrix $I_{m}$ zu einer $(m, n+m)$-Matrix $\tilde{A}=\left(A, I_{m}\right)$. Mit dem Vektor der Struktur- und Schlupfvariablen $x=\left(x_{1}, \ldots, x_{n}, x_{n+1}, \ldots, x_{n+m}\right)^{\top} \in \mathbb{R}^{n+m}$ lassen sich die Gleichungsrestriktionen von $P=$ dadurch kurz als $\widetilde{A} x=b$ schreiben, und die Nichtnegativitätsbedingungen werden zu $x \geq 0$. Um auch die Zielfunktion dazu passend schreiben zu können, setzen wir $c=\left(c_{1}, \ldots, c_{n}, 0, \ldots, 0\right)^{\top} \in \mathbb{R}^{n+m}$. Damit lässt sich $P_{=}$in der Form
$$\max c^{\boldsymbol{\top}} x \quad \text { s.t. } \tilde{A} x=b, x \geq 0$$
schreiben. Da sie die $m$ linear unabhängigen Spalten der Einheitsmatrix enthält, besitzt die erweiterte Koeffizientenmatrix $\widetilde{A}=\left(A, I_{m}\right)$ den Rang $m$. Wenn wir von der speziellen Struktur der Koeffizientenmatrix absehen und nur fordern, dass $\tilde{A}$ eine $(m, m+n)$-Matrix vom vollen Rang $m$ ist, dann bezeichnen wir die Form
$P_{\text {norm }}: \quad \max c^{\top} x \quad$ s.t. $\quad \widetilde{A} x=b, x \geq 0$
als Normalform eines linearen Optimierungsproblems. Falls die obige speziellere Struktur vorliegt, also $\widetilde{A}=\left(A, I_{m}\right)$ und $c=\left(c_{1}, \ldots, c_{n}, 0, \ldots, 0\right)^{\top}$ gelten, und wenn die zusätzliche Bedingung $b \geq 0$ erfüllt ist, dann liegt das lineare Optimierungsproblem in kanonischer Form vor.

## 统计代写|运筹学作业代写operational research代考|Zulässige Basislösung, Basis- und Nichtbasisvariablen

Eine Lösung $x=\left(x_{1}, \ldots, x_{n+m}\right)^{\top}$ der Restriktionen $\tilde{A} x=b$ eines linearen Optimierungsproblems in Normalform heißt Basislösung, wenn $n$ der Einträge $x_{i}$ von $x$ den Wert null haben und wenn die zu den restlichen $m$ Einträgen gehörenden Spalten $a^{i}$ von $\tilde{A}$ linear unabhängig sind. Die Bezeichnung als Basislösung liegt darin begründet, dass diese linear unabhängigen Spalten eine Basis des $\mathbb{R}^{m}$ bilden. Wenn die von null verschiedenen Einträge von $x$ außerdem nichtnegativ sind,sprechen wir von einer zulässigen Basislösung. Die $m$ linear unabhängigen Vektoren $a^{i}$ einer Basislösung nennt man Basisvektoren und die $m$ zugehörigen $x_{i}$ Basisvariablen oder kurz BV. Die $n$ verschwindenden Einträge $x_{i}$ von $x$ heißen entsprechend Nichtbasisvariablen oder kurz NBV, und die zugehörigen Vektoren $a^{i}$ Nichtbasisvektoren.

Im Folgenden fassen wir die Basisvektoren $a^{i}$ einer Basislösung $x$ zu der $(m, m)$ Matrix $B$ zusammen und die Nichtbasisvektoren zu der $(m, n)$-Matrix $N$. Mit derselben Indexsortierung spalten wir den Vektor $x$ in den Vektor der Basisvariablen $x_{B}$ und den Vektor der Nichtbasisvariablen $x_{N}$ auf. Das Gleichungssystem $\tilde{A} x=b$ lässt sich damit als
$$B x_{B}+N x_{N}=b$$
schreiben.
Da $B$ als quadratische Matrix mit linear unabhängigen Spalten invertierbar ist, lässt sich dieses System äquivalent zu
$$x_{B}+B^{-1} N x_{N}=B^{-1} b$$
umformen, also zu einem System mit Koeffizientenmatrix $\left(I_{m}, B^{-1} N\right)$ anstelle von $\widetilde{A}$ und rechter Seite $B^{-1} b$ anstelle von $b$. Durch diese Äquivalenzumformung kann man immer erreichen, dass jede der $m$ Basisvariablen in genau einer der $m$ Gleichungen vorkommt, und dies sogar mit dem Koeffizienten eins. Ferner liest man sofort ab, dass die Basislösung durch $x_{N}=0$ und $x_{B}=B^{-1} b$ gegeben ist.

Der Simplex-Algorithmus zeichnet sich unter anderem dadurch aus, dass die aufwändige Berechnung von $B^{-1}$ zur Bestimmung einer Basislösung dadurch umgangen wird, dass $B^{-1}$ als effizient auszuführender Update der entsprechenden inversen Basismatrix der vorhergehenden Basislösung ermittelt wird.

## 统计代写|运筹学作业代写operational research代考|Normalform und kanonische Form

$$\max c^{\top} x \quad \text { s.t. } \tilde{A} x=b, x \geq 0$$

$P_{\text {norm }}: \max c^{\top} x \quad$ 英石 $\widetilde{A} x=b, x \geq 0$

## 统计代写|运筹学作业代写operational research代考|Zulässige Basislösung, Basis- und Nichtbasisvariablen

$$B x_{B}+N x_{N}=b$$

$$x_{B}+B^{-1} N x_{N}=B^{-1} b$$

## 有限元方法代写

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## 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。