分类： 商科代写

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|商业分析作业代写Statistical Modelling for Business代考|The Data Component

A common starting point for a discussion about data is that they are facts. There is a huge philosophical literature on what is a fact. As noted by Mulligan and Correia (2020), a fact is the opposite of theories and values and “are to be distinguished from things, in particular from complex objects, complexes and wholes, and from relations.” Without getting into this philosophy of facts, I will hold that a fact is a checkable or provable entity and, therefore, true. For example, it is true that Washington D.C. is the capital of the United States: it is easily checkable and can be shown to be true. It is also a fact that $1+1=2$. This is checkable by simply counting one finger on your left hand and one finger on your right hand. ${ }^3$

You could have a lot of facts on a topic but they are of little value if they are not

1. organized,
2. subsetted,
3. manipulated, and
4. interpreted
in a meaningfully way to provide insight for a recommendation for an action, the action being the problem solution. Otherwise, the facts are just a collection of valueless things. Their value stems from what you can do with them.

Organizing data, or facts, is a first step in any analytical process and the drive for information. This could involve arranging them in chronological order (e.g., by date and time of a transaction), spatial order (e.g., countries in the Northern and Southern Hemispheres), alphanumeric order, size order, and so on in an infinite number of ways. Transactions data, for example, are facts about units sold of a series of products including what products, who bought them, when they were sold, the amount sold, and prices. They are typically maintained in a file without a discernible order: just product, date, and units. There is no insight or intelligence from this data. In fact, it is somewhat randomly organized based on when orders were placed. ${ }^4$ If sorted by product and date, however, then they are organized and useful, but not much. The best organization is the one most applicable for a practical problem.
In Data Science, statistics, econometrics, machine learning, and other quantitative areas, a common organizational form is a rectangular array consisting of rows and columns: The rows are typically objects and the columns variables or features. An object can be a person (e.g., a customer) or an event (e.g., a transaction). The words object, case, individual, event, observation, and instance are often used interchangeably and I will certainly do this. For the methods considered in this book, each row is an individual case, one case per row and each case is in its own row.

商科代写|商业分析作业代写Statistical Modelling for Business代考|The Extractor Component

Finally, you have to apply some methods or procedures to your DataFrame to extract information. Refer back to Fig. $1.2$ for the role and position of an Extractor function in the information chain. This whole book is concerned with these methods. The interpretation of the results to give meaning to the information will be illustrated as I develop and discuss the methods, but the final interpretation is up to you based on your problem, your domain knowledge, and your expertise.

Due to the size and complexity of modern business data sets, the amount and type of information hidden inside them is large, to say the least. There is no one piece of information-no one size fits all-for all business problems. The same data set can be used for multiple problems and can yield multiple types of information. The possibilities are endless. The information content, however, is a function of the size and complexity of the DataFrame you eventually work with. The size is the number of data elements. Since a DataFrame is a rectangular array, the size is #rows $\times$ #columns elements and is given by its shape attribute. Shape is expressed as a tuple written as (rows, columns). For example, it could be $(5,2)$ for a DataFrame with 5 rows and 2 columns and 10 elements. The complexity is the types of data in the DataFrame and is difficult to quantify except to count types. They could be floating point numbers (or, simply, floats), integers (or ints), Booleans, text strings (referred to as objects), datetime values, and more. The larger and more complex the DataFrame, the more information you can extract. Let $I=I$ fformation, $S=$ size and $C=$ complexity. Then Information $=f(S, C)$ with $\partial I / \partial S>0$ and $\partial I / \partial C>0$. For a very large, complex DataFrame, there is a very large amount of information.

The cost of extracting information directly increases with the DataFrame’s size and complexity of the data. If I have 10 sales values, then my data set is small and simple. Minimal information, such as the mean, standard deviation, and range, can be extracted. The cost of extraction is low; just a hand-held calculator is needed. If I have $10 \mathrm{~GB}$ of data, then more can be done but at a greater cost. For data sizes approaching exabytes, the costs are monumental.

There could be an infinite amount of information, even contradictory information, in a large and complex DataFrame. So, when something is extracted, you have to check for its accuracy. For example, suppose you extract information that classifies customers by risk of default on extended credit. This particular classification may not be a good or correct one; that is, the predictive classifier $(P C)$ may not be the best one for determining whether someone is a credit risk or not. Predictive Error Analysis $(P E A)$ is needed to determine how well the $P C$ worked. I discuss this in Chap. 11. In that discussion, I will use a distinction between a training data set and a testing data set for building the classifier and testing it. This means the entire DataFrame will not, and should not, be used for a particular problem. It should be divided into the two parts although that division is not always clear, or even feasible. I will discuss the split into training and testing data sets in Chap.9.
The complexity of the DataFrame is, as I mentioned above, dependent on the types of data. Generally speaking, there are two forms: text and numeric. Other forms such as images and audio are possible but I will restrict myself to these two forms. I have to discuss these data types so that you know the possibilities and their importance and implications. How the two are handled within a DataFrame and with what statistical, econometric, and machine learning tools for the extraction of information is my focus in this book and so I will deal with them in depth in succeeding chapters. I will first discuss text data and then numeric data in the next two subsections.

商业分析代写

商科代写|商业分析作业代写Statistical Modelling for Business代考|The Data Component

1. 有组织的，
2. 子集，
3. 操纵，和
4. 以有意义的方式进行解释
，以提供对行动建议的洞察力，行动就是问题的解决方案。否则，事实只是一堆毫无价值的东西。它们的价值源于您可以用它们做什么。

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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|商业分析作业代写Statistical Modelling for Business代考|Uncertainty vs. Risk

Uncertainty is a fact of life reflecting our lack of knowledge. It is either spatial (“I don’t know what is happening in Congress today.”) or temporal (“I don’t know what will happen to sales next year.”). In either case, the lack of knowledge is about the state of the world (SOW): what is happening in Congress and what will happen next year. Business textbooks such as Freund and Williams (1969), Spurr and Bonini (1968), and Hildebrand et al. (2005) typically discuss assigning a probability to different $S O W$ s that you could list. The purpose of these probabilities is to enable you to say something about the world before that something materializes. Somehow, and it is never explained how, you assign numeric values representing outcomes, or payoffs, to the $S O W \mathrm{~s}$. The probabilities and associated payoffs are used to calculate an expected or average payoff over all the possible $S O W$ s. Consider, for example, the rate of return on an investment (ROI) in a capital expansion project. The ROI might depend on the average annual growth of real GDP for the next 5 years. Suppose the real GDP growth is simply expressed as declining (i.e., a recession), flat ( $0 \%$ ), slow $(1 \%-2 \%)$, and robust $(>2 \%)$ with assigned probabilities of $0.05,0.20,0.50$, and $0.25$, respectively. These form a probability distribution. Let $p_i$ be the probability state $i$ is realized. Then, $\sum_{i=1}^n p_i=1.0$ for these $n=4$ possible states. I show the $S O W \mathrm{~s}$, probabilities, and $R O I$ values in Table 1.1. The expected $R O I$ is $\sum_{i=1}^4 p_i \times$ $R O I_i=2.15 \%$. This is the amount expected to be earned on average over the next 5 years.

Savage (1972, p. 9) notes that the “world” in the statement “state of the world” is defined for the problem at hand and that you should not take it literally. It is a fluid concept. He states that it is “the object about which the person is concerned.” At the same time, the “state” of the world is a full description of its conditions. Savage (1972) notes that it is “a description of the world, leaving no relevant aspects undescribed.” But he also notes that there is a true state, a “state that does in fact obtain, i.e., the true description of the world.” Unfortunately, it is unknown, and so the best we can do until it is realized or revealed to us is assign probabilities to the occurrence of each state for decision making. These are the probabilities in Table 1.1. More importantly, it is the fact that the true state is unknown, and never will be known until revealed that is the problem. No amount of information will ever completely and perfectly reveal this true state before it occurs.

商科代写|商业分析作业代写Statistical Modelling for Business代考|The Data-Information Nexus

To an extent, discussing definitions and terminology is useful for the advancement of scientific and practical solutions for any problem. If you cannot agree on basic terms, then you are doomed at worst and hindered at best from making any progress toward a solution, a decision. You can, however, become so involved in defining terms and so overly concerned about terminology that nothing else maters. Popper too strongly, that
One should never quarrel about words, and never get involved in questions of terminology … What we are really interested in, our real problem,… are problems of theories and their truth.
Popper, a philosopher of science, was concerned about scientific problems. The same sentiment, however, holds for practical problems like the ones you face daily in your business. Despite Popper’s preeminence, you still need some perspective on the foundational units that drive the raison d’etre of BDA: data and information. ${ }^1$ If information is so important for reducing uncertainty, then a logical question to ask is: “What is information?” A subordinate, but equally important, question is:

The words information and data are used as synonyms in everyday conversations. It is not uncommon, for example, to hear a business manager claim in one instance that she has a lot of data and then say in the next instance that she has a lot of information, thus linking the two words to have the same meaning. In fact, the computer systems that manage data are referred to as Information Systems (IS) and the associated technology used in those systems is referred to as Information Technology (IT). ${ }^2$ The C-Level executive in charge of this data and $I T$ infrastructure is the Chief Information Officer $(\mathrm{CIO})$. Notice the repeated use of the word “information.”
Even though people use these two words interchangeably it does not mean they have the same meaning. It is my contention, along with others, that data and information are distinct terms that, yet, have a connection. I will simply state that data are facts, objects that are true on their face, that have to be organized and manipulated to yield insight into something previously unknown. When managed and manipulated, they become information. The organization cannot be without the manipulation and the manipulation cannot be without the organization. The IT group of your business organizes your company’s data but it does not manipulate it to be information. The information is latent, hidden inside the data and must be extracted so it can be used in a decision. I illustrate this connection Fig. 1.2. I will comment on each component in the next few sections.

商业分析代写

商科代写|商业分析作业代写Statistical Modelling for Business代考|Uncertainty vs. Risk

Savage (1972, p. 9) 指出，“世界状况”陈述中的“世界”是为手头的问题定义的，你不应该从字面上理解它。这是一个流动的概念。他说这是“这个人所关心的对象”。同时，世界的“状态”是对其状况的完整描述。Savage (1972) 指出它是“对世界的描述，没有留下任何未描述的相关方面”。但他也指出存在一种真实的状态，一种“确实获得的状态，即对世界的真实描述”。不幸的是，它是未知的，因此在它被实现或揭示给我们之前我们能做的最好的事情就是为每个状态的发生分配概率以进行决策。这些是表 1.1 中的概率。更重要的是，真实状态不明，在发现问题所在之前永远不会为人所知。在这种真实状态发生之前，再多的信息也无法完全、完美地揭示它。

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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

What types of business problems warrant $B D A$ ? The types are too numerous to mention, but to give a sense of them consider a few examples:

• Anomaly Detection: production surveillance, predictive maintenance, manufacturing yield optimization;
• Fraud detection;
• Identity theft;
• Account and transaction anomalies;
• Customer analytics:
• Customer Relationship Management (CRM);
• Churn analysis and prevention;
• Customer Satisfaction;
• Marketing cross-sell and up-sell;
• Pricing: leakage monitoring, promotional effects tracking, competitive price responses;
• Fulfillment: management and pipeline tracking;
• Competitive monitoring;
• Competitive Environment Analysis (CEA); and
• New Product Development.
And the list goes on, and on.
A decision of some type is required for all these problems. New product development best exemplifies a complex decision process. Decisions are made throughout a product development pipeline. This is a series of stages from ideation or conceptualization to product launch and post-launch tracking. Paczkowski (2020) identifies five stages for a pipeline: ideation, design, testing, launch, and post-launch tracking. Decisions are made between each stage whether to proceed to the next one or abort development or even production. Each decision point is marked by a business case analysis that examines the expected revenue and market share for the product. Expected sales, anticipated price points (which are refined as the product moves through the pipeline), production and marketing cost estimates, and competitive analyses that include current products, sales, pricing, and promotions plus competitive responses to the proposed new product, are all needed for each business case assessment. If any of these has a negative implication for the concept, then it will be canceled and removed from the pipeline. Information is needed for each business case check point.

The expected revenue and market share are refined for each business case analysis as new and better information -not data-become available for the items I listed above. More data do become available, of course, as the product is developed, but it is the analysis of that data based on methods described in this book, that provide the information needed to approve or not approve the advancement of the concept to the next stage in the pipeline. The first decision, for example, is simply to begin developing a new product. Someone has to say “Yes” to the question “Should we develop a new product?” The business case analysis provides that decision maker with the information for this initial “Go/No Go” decision. Similar decisions are made at other stages.

Another example is product pricing. This is actually a two-fold decision involving a structure (e.g., uniform pricing or price discrimination to mention two possibilities) and a level within the structure. These decisions are made throughout the product life cycle beginning at the development stage (the launch stage of the pipeline I discussed above) and then throughout the post-launch period until the product is ultimately removed from the market. The wrong price structure and/or level could cost your business lost profit, lost market share, or a lost business. See Paczkowski (2018) for a discussion of the role of pricing and the types of analysis for identifying the best price structure and level. Also see Paczkowski (2020) for new product development pricing at each stage of the pipeline.

Decisions are effective if they solve a problem, such as those I discussed above, and aid rather than hinder your business in succeeding in the market. I will assume your business succeeds if it earns a profit and has a positive return for its owners (shareholders, partners, employees in an employee-owned company) or a sole owner. Information could be about

• current sales;
• future sales;
• the state of the market;
• consumer, social, and technology trends and developments;
• customer needs and wants;
• customer willingness-to-pay;
• key customer segments;
• financial developments;
• supply chain developments; and
• the size of customer churn.
This information is input into decisions and like any input, if it is bad, then the decisions will be bad. Basically, the GIGO Principle (Garbage In-Garbage Out) holds. This should be obvious and almost trite. Unfortunately, you do not know when you make your decision if your information is good or bad, or even sufficient. You face uncertainty due to the amount and quality of the information you have available.

Without any information you would just be guessing, and guessing is costly. In Fig. 1.1, I illustrate what happens to the cost of decisions based on the amount of information you have. Without any information, all your decisions are based on pure guesses, hunches, so you are forced to approximate their effect. The approximation could be very naive, based on gut instinct (i.e., an unfounded belief that you know everything) or what happened yesterday or in another business similar to yours (i.e., an analog business).

The cost of these approximations in terms of financial losses, lost market share, or outright bankruptcy can be very high. As the amount of information increases, however, you will have more insight so your approximations (i.e., guesses) improve and the cost of approximations declines. This is exactly what happens during the business case process I described above. More and better information helps the decision makers at each business case stage. The approximations could now be based on trends, statistically significant estimates of impact, or model-based what-if analyses. These are not “data”; they are information.

商业分析代写

• 异常检测：生产监控、预测性维护、制造良率优化；
• 欺诈识别;
• 身份盗用；
• 账户及交易异常；
• 客户分析：
• 客户关系管理（CRM）；
• 客户流失分析与预防；
• 顾客满意度;
• 营销交叉销售和追加销售；
• 定价：泄漏监控、促销效果跟踪、有竞争力的价格响应；
• 履行：管理和管道跟踪；
• 竞争监控；
• 竞争环境分析（CEA）；和
• 新产品开发。
这样的例子不胜枚举。
所有这些问题都需要某种类型的决定。新产品开发最能说明复杂的决策过程。决策是在整个产品开发流程中做出的。这是从构思或概念化到产品发布和发布后跟踪的一系列阶段。Paczkowski (2020) 确定了管道的五个阶段：构思、设计、测试、发布和发布后跟踪。在每个阶段之间做出决定是继续下一阶段还是中止开发甚至生产。每个决策点都由业务案例分析标记，该分析检查产品的预期收入和市场份额。预期销售额、预期价格点（随着产品在管道中移动而细化）、生产和营销成本估算以及包括当前产品的竞争分析，每个业务案例评估都需要销售、定价和促销以及对拟议新产品的竞争性反应。如果其中任何一个对该概念有负面影响，那么它将被取消并从管道中删除。每个业务案例检查点都需要信息。

• 当前销售额；
• 未来的销售；
• 市场状况；
• 消费者、社会和技术趋势和发展；
• 客户的需求和愿望；
• 客户支付意愿；
• 关键客户群；
• 金融发展；
• 供应链发展；和
• 客户流失的规模。
这些信息被输入到决策中，就像任何输入一样，如果它是错误的，那么决策就会是错误的。基本上，GIGO 原则（垃圾进垃圾出）成立。这应该是显而易见的，几乎是陈腐的。不幸的是，您不知道您何时做出决定，您的信息是好是坏，甚至是充分的。由于可用信息的数量和质量，您面临着不确定性。

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。

商科代写|计量经济学代写Econometrics代考|Best 27

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics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。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|计量经济学代写Econometrics代考|Formal Tests for Serial Correlation

The Durbin-Watson (DW) test provides a formal test in which the null hypothesis is that the equation errors are serially uncorrelated and the alternative is that they follow a first-order autocorrelation process. This test was first introduced by Durbin and Watson in two papers published in Biometrika in 1950 and 1951 [Durbin1950] and [Durbin1951]. It is a standard part of the regression output for most econometrics packages. The DW test builds on a previous test developed by Von Neumann [VonNeumann1941] who developed a test for autocorrelation in a series of random variables with the null that the variables are independent random numbers. Unfortunately, this is not suitable when the series under examination comprises regression residuals, which are not independent by construction. Although Von Neumann’s statistic has a relatively simple distribution, that is, the standard normal distribution, Durbin and Watson showed that the distribution of their test statistic was necessarily more complex. The nature of the test statistic means that it is not possible to derive unique critical values for a test of the null of no autocorrelation against the alternative of first-order autocorrelation. However, they did demonstrate that the critical values for their test were bounded and were able to tabulate the bounds for small sample sizes.
The DW test is concerned with a specific form of serial correlation, that is, first-order autocorrelation but is arguably sensitive to other forms. Consider the following regression model with an error that follows an AR process of order one:
\begin{aligned} &Y_{t}=\beta X_{t}+u_{t} \ &u_{t}=\rho u_{t-1}+\varepsilon_{t} . \end{aligned}
Taking the residuals from an OLS regression of $Y$ on $X$, we can construct the test statistic
$$D W=\sum_{t=2}^{T}\left(\hat{u}{t}-\hat{u}{t-1}\right)^{2} / \sum_{t=1}^{T} \hat{u}_{t}^{2} .$$

商科代写|计量经济学代写Econometrics代考|DEALING WITH SERIAL CORRELATION

If serial correlation is present, then there are several ways to deal with the issue. Of course, the priority is to identify the nature, and hopefully the cause, of the serial correlation. If the root cause of the problem is the omission of a relevant variable from the model, then the natural solution is to include that variable. If it is determined that modeling of the serial correlation process is appropriate, then we have several different methods available for the estimation of such models by adjusting for the presence of serially correlation errors. It should be noted that mechanical adjustments, of the type we will describe in this section, are potentially dangerous. This process has been much criticized on the grounds that there is a risk that these methods disguise an underlying problem rather than dealing with it. McGuirk and Spanos [McGuirk2009] are particularly critical of mechanical adjustments to deal with autocorrelated arguments. In this paper, they show that unless we can assume that the regress and does not Granger-cause the regressors, adjusting for autocorrelation means that least squares yield biased and inconsistent estimates. However, these methods are still used and reported in applied work and it is therefore important that we consider how they work.

The first method we will consider is that of Cochrane-Orcutt estimation. This uses an iterative algorithm proposed by Cochrane and Orcutt [Cochrane1949] in which we use the structure of the problem to separate out the estimation of the behavioral parameters of the main equation from those of the AR process that describes the errors. Let us consider the case of an $\mathrm{AR}(1)$ error process as an example. Suppose we wish to estimate a model of the form (5.6). The two equations can be combined to give a single equation of the form
$$Y_{t}-\rho Y_{t-1}=\beta\left(X_{t}-\rho X_{t-1}\right)+\varepsilon_{t},$$
that is, an equation in “quasi-differences” of the data. If $\rho$ was known, then it would be straightforward to construct these quasi-differences and estimate the behavioral parameter $\beta$ by least squares. In the absence of such knowledge, we make a guess at $\rho$ and construct an estimate of $\beta$ on this basis. We then generate the residuals $\hat{u}{t}=Y{t}-\beta X_{t}$ on this basis and calculate an estimate of $\rho$ of the form $\hat{\rho}=\sum_{t=2}^{T} \hat{u}{t} \hat{u}{t-1} / \sum_{t=1}^{T} \hat{u}_{t}^{2}$. If, by some lucky chance, this estimate coincides with our assumption, then we stop. Otherwise, we use our estimate to recalculate the quasi-differences, reestimate $\beta$, and continue until our estimates of $\beta$ and $p$ converge. If a solution exists, then this provides a robust algorithm for estimation.

商科代写|计量经济学代写Econometrics代考|Formal Tests for Serial Correlation

Durbin-Watson (DW) 检验提供了一种形式检验，其中零假设是方程误差是序列不相关的，而另一种方法是它们 遵循一阶自相关过程。该测试由 Durbin 和 Watson 在 1950 年和 1951 年在 Biometrika 发表的两篇论文 [Durbin1950] 和 [Durbin1951] 中首次引入。它是大多数计量经济学软件包回归输出的标准部分。DW 测试建立 在 Von Neumann [VonNeumann1941] 开发的先前测试的基础上，该测试开发了一系列随机变量的自相关测 试，其中变量为独立随机数。不幸的是，当检查的序列包含回归残差时，这不适合，这些回归残差在构造上不是 独立的。尽管冯诺依曼的统计量具有相对简单的分布，即标准正态分布，但 Durbin 和 Watson 表明，他们的检 验统计量的分布必然更复杂。检验统计量的性质意味着不可能为无自相关的零点与一阶自相关的备选方案的检验 推导出唯一的临界值。然而，他们确实证明了他们测试的临界值是有界的，并且能够将小样本的界限制表。检验 统计量的性质意味着不可能为无自相关的零点与一阶自相关的备选方案的检验推导出唯一的临界值。然而，他们 确实证明了他们测试的临界值是有界的，并且能够将小样本的界限制表。检验统计量的性质意味着不可能为无自 相关的零点与一阶自相关的备选方案的检验推导出唯一的临界值。然而，他们确实证明了他们测试的临界值是有 界的，并且能够将小样本的界限制表。

DW 检验关注特定形式的序列相关，即一阶自相关，但可以说对其他形式敏感。考虑以下回归模型，其误差遵循 一阶 AR 过程:
$$Y_{t}=\beta X_{t}+u_{t} \quad u_{t}=\rho u_{t-1}+\varepsilon_{t} .$$

$$D W=\sum_{t=2}^{T}(\hat{u} t-\hat{u} t-1)^{2} / \sum_{t=1}^{T} \hat{u}_{t}^{2} .$$

商科代写|计量经济学代写Econometrics代考|DEALING WITH SERIAL CORRELATION

$$Y_{t}-\rho Y_{t-1}=\beta\left(X_{t}-\rho X_{t-1}\right)+\varepsilon_{t},$$

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。

商科代写|计量经济学代写Econometrics代考|Best 22

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics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。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|计量经济学代写Econometrics代考|DETECTION OF SERIAL CORRELATION

Since serial correlation of the errors has been shown to have important implications for our interpretation of regression results, it becomes important to develop tests for whether it is present in the models we estimate. It will be helpful to have an example in mind as we develop such tests. Consider the following estimated consumption function for the US economy based on annual data from 1970 to $2019 .$
$\ln \left(C_{t}\right)=\underset{(0.0466)}{-0.2898}+\underset{(0.0052)}{1.0174} \ln \left(Y D_{t}\right)+\hat{u}{t}$ $R^{2}=0.9987 \quad \hat{\sigma}{u}=0.0154 \quad T=50$
$D W=0.5389$,
where $C$ is total consumers’ expenditure and $Y D$ is real personal disposable income. Both variables are measured in millions of dollars at 2012 prices.
On first inspection, this equation appears to have reasonable properties. The slope coefficient is statistically significantly different from zero and the coefficient of determination indicates a good fit. In this case, however, it is more interesting to test the null hypothesis that the slope coefficient is equal to 1 . This is because the slope coefficient in this relationship measures the income elasticity of consumption expenditure. However, this null hypothesis is also rejected by our model. The $t$ statistic for this test is $(1.0174-1) / 0.0052=3.35$ which leads us to reject the null at the $5 \%$ level. This conclusion is, however, may be unreliable if the errors in this model are serially correlated. Therefore, in order to assess the robustness of our estimates of the model parameters, we need to test for the presence of serial correlation in the residuals $\hat{u}$ from equation (5.18).

商科代写|计量经济学代写Econometrics代考|Informal Tests for Serial Correlation

We can look for serial correlation informally by simply inspecting a plot of the residuals. If runs of positive or negative residuals are obvious, then this is a sign that serial correlation is present. For example, Figure $5.2$ shows the residuals for our consumption function equation and it is obvious that there are periods during which the residuals are consistently either positive or negative. This is a clear indication that the equation suffers from serial correlation. There are, however, forms of serial correlation, such as moving average errors, which are not so easily detected. Therefore, it is important to develop more formal tests as well as procedures for identifying the specific form of serial correlation that is relevant for this equation.

The correlogram 1 provides a more formal statistical method for the investigation of serial correlation. The correlogram is a table, or plot, of the sample autocorrelations of the regression residuals. The sample autocorrelations are defined in equation (5.19), and Figure $5.3$ gives both a table and a graph of the autocorrelations of the residuals from equation (5.18).
$$\hat{\rho}{k}=\sum{t=k+1}^{T} \hat{u}{t} \hat{u}{t-k} / \sum_{t=1}^{T} \hat{u}_{t^{2}}^{2} .$$

商科代写|计量经济学代写Econometrics代考|Informal Tests for Serial Correlation

$$\hat{\rho} k=\sum t=k+1^{T} \hat{u} t \hat{u} t-k / \sum_{t=1}^{T} \hat{u}_{t^{2}}^{2} .$$

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。

商科代写|计量经济学代写Econometrics代考|Find 2022

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics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。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|计量经济学代写Econometrics代考|CAUSES OF SERIAL CORRELATION

Consider a regression equation of the standard form $Y_{t}=\beta X_{t}+u_{t}$. We assume a model in mean deviation form to simplify the notation and the discussion. The errors of this model are said to be serially correlated if $E\left(u_{t} u_{t-k}\right) \neq 0$ for some $k \neq 0$. A natural question is why the errors might be correlated in this way? For the moment, we will simply assume that this is an intrinsic property of the data. That is, we assume that shocks to the equation are not random drawings from a distribution but instead depend upon their own past values. An alternative would be to assume that correlation in the errors arises because the model is misspecified in some way. However, this would complicate much of the discussion and we will avoid this assumption for the moment, on the understanding that it will be relaxed later.

There are many different forms that serial correlation might take. For example, the errors might follow a first-order autoregressive (AR) process. This would mean that the error process could be described by an equation of the form $u_{t}=\rho u_{t-1}+\varepsilon_{t}$, where $\varepsilon_{t}$, is a truly random disturbance and $\rho \neq 0$. This is a very common and important case, but it is not the only form that serial correlation can take. An alternative is where the error term in the equation is an average over several time periods of the random disturbance $\varepsilon_{t}$. For example, we might have a first-order moving average process of the form $u_{t}=\varepsilon_{t}+\lambda \varepsilon_{t-1}$. Both error processes are said to be serially correlated but each produces different implications and problems for the modeler. However, in both cases, the problem of dealing with serial correlation is simplified because of the assumption that it is an intrinsic feature of the error themselves, that is, the problem is one of error dynamics. A more realistic conclusion might be that the errors are serially correlated because of some fundamental misspecification in the original equation.

商科代写|计量经济学代写Econometrics代考|CONSEQUENCES OF SERIAL CORRELATION

Now that we have established some of the reasons why serial correlation may arise in regression models, let us consider the implications for least squares regression analysis. Suppose we have a model in which the errors follow a first-order AR process as set out in (5.6)
\begin{aligned} &Y_{t}=\beta X_{t}+u_{t} \ &u_{t}=\rho u_{t-1}+\varepsilon_{t}, \end{aligned}
where $\varepsilon_{t}, t=1, \ldots, T$ are independent, identically distributed random disturbances with mean zero and constant variance. As we have seen, this is not the only possible type of serial correlation which may arise, but the results we derive for this model apply more generally to other forms of serial correlation.

The AR process defined in (5.6) can be written in moving average form. Using the method of backward substitution, we have
$$u_{t}=\varepsilon_{t}+\rho \varepsilon_{t-1}+\rho^{2} \varepsilon_{t-2} \ldots=\sum_{j=0}^{\infty} \rho^{j} \varepsilon_{t-j} .$$
This is an infinite moving average process. Providing $|\rho|<1$, then the sequence defined in (5.7) will converge, in the sense that it will have a finite variance. To see this note that
$$E\left(u_{t}^{2}\right)=\sum_{j=0}^{\infty} \rho^{2 j} E\left(\varepsilon_{t-j}^{2}\right)=\sum_{j=0}^{\mu_{j}} \rho^{2 j} \sigma_{\varepsilon}^{2}=\frac{\sigma_{\varepsilon}^{2}}{1-\rho^{2}}$$
Therefore, for the variance of the error term to be finite and positive, we need $|\rho|<1$. If this condition holds, then the process is said to be weakly stationary and it can be shown that a general feature of stationary, finite AR processes is that they can be written as infinite moving average processes. Moreover, since $E\left(\varepsilon_{t-j}\right)=0$ for all values of $j$, it follows that $E\left(u_{t}\right)=0$. This is a useful property because we have already seen that the expected value of the OLS estimator can be written as follows: $E(\hat{\beta})=\beta+\sum_{t=1}^{T} X_{t} E\left(u_{t}\right) / \sum_{t=1}^{T} X_{t}^{2}$. It therefore follows that $E(\hat{\beta})=\beta$ and that the OLS estimator is unbiased even when the errors are serially correlated.

商科代写|计量经济学代写Econometrics代考|CONSEQUENCES OF SERIAL CORRELATION

$$Y_{t}=\beta X_{t}+u_{t} \quad u_{t}=\rho u_{t-1}+\varepsilon_{t},$$

(5.6) 中定义的 AR 过程可以写成移动平均形式。使用向后替换的方法，我们有
$$u_{t}=\varepsilon_{t}+\rho \varepsilon_{t-1}+\rho^{2} \varepsilon_{t-2} \ldots=\sum_{j=0}^{\infty} \rho^{j} \varepsilon_{t-j} .$$

$$E\left(u_{t}^{2}\right)=\sum_{j=0}^{\infty} \rho^{2 j} E\left(\varepsilon_{t-j}^{2}\right)=\sum_{j=0}^{\mu_{j}} \rho^{2 j} \sigma_{\varepsilon}^{2}=\frac{\sigma_{\varepsilon}^{2}}{1-\rho^{2}}$$

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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

First, let us discuss games of chance. Games of chance include, but are not limited to, games involving flipping coin, rolling dice, drawing cards from a deck, spinning a wheel, and such. Suppose you are flipping a coin with a friend. If you both flip the same, both heads or both tails, you win $\$ 1$. If you flip a head and a tail each, you lose$\$1$. If you play this game with this bet 100 times over the course of the evening, how much do you expect to win? Or lose? To answer such questions, we need two concepts: the probability of an event and the expected value of a random variable.

In situations like this, it makes sense to count the number of times something occurs. An efficient way to do this is to use the frequency definition of the probability of an event. The probability of the event two heads or two tails is the number of ways we can achieve these results divided by the total number of possible outcomes. That is, we define both coins being flipped that land on both heads or both tails as a favorable event $A$. We can define event $A$ as the set of outcomes that include ${\mathrm{HH}, \mathrm{TT}}$.
Favorable outcomes are those that consist of ${\mathrm{HH}, \mathrm{TT}}$.
$$\begin{array}{r} \text { Probability of an event }=\frac{\text { Favorable outcomes }}{\text { Total outcomes }} \ \text { Probability of an event }{A}=\frac{\text { Number of outcomes of }{A}}{\text { Total outcomes }} \end{array}$$

Of course, the probability of an event (flip of a fair coin) must be equal or greater than zero, and equal to or less than 1. And the sum of the probabilities of all possible events must equal 1 . That is,
\begin{aligned} &0 \leq p_{i} \leq 1 \ &\sum_{i=1}^{n} p_{i}=1, i=1,2, \ldots, n \end{aligned}
We need to compute all the possible outcomes of flipping two coins, and then determine how many result in the same results defined by event $A$. A tree is useful for visualizing the outcomes. These outcomes constitute the sample space. On the first flip, the possible results are $\mathrm{H}$ or $\mathrm{T}$. And on the second flip, the same outcomes are still available. We assume that these events are equally likely to occur based on flipping and obtaining either a head or tail of each flip.

First, we define a random variable as a rule that assigns a number to every outcome of a sample.

We use $E[X]$, which is stated as the expected value of $X$. We define $E[X]$ as follows:
Expected value, $E[X]$, is the mean or average value.
Further, we provide the following two formulas: in the discrete case, $E[X]=\sum_{i=1}^{n} x_{i} p\left(x_{i}\right)$ and in the continuous case, $E[X]=\int_{-\infty}^{+\infty} x * f(x) d x$.

There are numerous ways to calculate the average value. We present a few common methods that you could use in decision theory.

If you had 2 quiz grades, an 82 and a 98 , almost intuitively you would add the two numbers and divide by 2 , giving an average of 90 .

Average scores: Two scores that were earned were 82 and 98 . Compute the average.
$$E[X]=\frac{(82+98)}{2}=90$$
If after 5 quizzes, you had three 82 s and two 98 s, you would add them and divide by $5 .$
$$\text { Average }=\frac{3(82)+2(98)}{5}=88.4$$
Rearranging the terms, we obtain
$$\text { Average }=\frac{3}{5}(82)+\frac{2}{5}(98)$$
In this form, we have two payoffs, 82 and 98 , each multiplied by the weights, $3 / 5$ and $2 / 5$. This is analogous to the definition of expected value.

Suppose a game has outcomes $a_{1}, a_{2}, \ldots, a_{n}$, each with a payoff $w_{1}, w_{2}, \ldots, w_{n}$ and a corresponding probability $p_{1}, p_{2}, \ldots, p_{n}$ where $p_{1}+p_{2}+\ldots+p_{n}=1$ and $0 \leq p_{i} \leq 1$, then the quantity
$$E=w_{1} p_{1}+w_{1} p_{2}+\ldots+w_{1} p_{n}$$
is the expected value of the game. Note that expected value is analogous to weighted average, but the weights must be probabilities $\left(0 \leq p_{i} \leq 1\right)$ and the weights must sum to 1 .

商业数学代考

Probability of an event $=\frac{\text { Favorable outcomes }}{\text { Total outcomes }}$ Probability of an event $A=\frac{\text { Number of outcomes of } A}{\text { Total outcomes }}$

$$0 \leq p_{i} \leq 1 \quad \sum_{i=1}^{n} p_{i}=1, i=1,2, \ldots, n$$

$$E[X]=\frac{(82+98)}{2}=90$$

$$\text { Average }=\frac{3(82)+2(98)}{5}=88.4$$

$$\text { Average }=\frac{3}{5}(82)+\frac{2}{5}(98)$$

$$E=w_{1} p_{1}+w_{1} p_{2}+\ldots+w_{1} p_{n}$$

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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

We will illustrate some mathematical models that describe change in the real world. We will solve some of these models and will analyze how good our resulting mathematical explanations and predictions are in context of the problem. The solution techniques that we employ take advantage of certain characteristics that the various models enjoy as realized through the formulation of the model.

When we observe change, we are often interested in understanding or explaining why or how a particular change occurs. Maybe we need or want to analyze the effects under different conditions or perhaps to predict what could happen in the future. Consider the firing of a weapon system or the shooting of a ball from a catapult as shown in Figure 1.3. Understanding how the system behaves in different environments under differing weather or operators, or predicting how well it hits the targets are all of interest. For the catapult, the critical elements of the ball, the tension, and angle of the firing arm are found as important elements (Fox, 2013b). For our purposes, we will consider a mathematical model to be a mathematical construct that is designed to study a particular real-world system or behavior (Giordano et al., 2014). The model allows us to use mathematical operations to reach mathematical conclusions about the model as illustrated in Figure 1.4. It is the arrow going from real-world system and observations to the mathematical model using the assumptions, variables, and formulations that are critical in the process.

We define a system as a set of objects joined by some regular interaction or interdependence in order for the complete system to work together. Think of a larger business with many companies that work independently and interact together to make the business prosper. Other examples might include a bass and trout population living in a lake, a communication, cable TV, or weather satellite orbiting the earth, delivering Amazon Prime packages, U.S. postal service mail or packages, locations of emergency services or computer terminals, or large companies’ online customer buying systems. The person modeling is interested in understanding how a system works, what causes change in a system, and the sensitivity of the system to change. Understanding all these elements will help in building an adequate model to replicate reality. The person modeling is also interested in predicting what changes might occur and when these changes might occur.

An outline is presented as a procedure to help construct mathematical models. In the next section, we will illustrate this procedure with a few examples. We suggest a nine-step process.

These nine steps are summarized in Figure 1.6. These steps act as a guide for thinking about the problem and getting started in the modeling process. We choose these steps from the compilation of steps by other authors listed in additional readings and put them together in these nine steps.

We illustrate the process through an example. Consider building a model where we want to identify the spread of a contagious disease.
Step 1: Understand the decision to be made, the question to be asked, or the problem to be solved.

Understanding the decision is the same as identifying the problem to be solved. Identifying the problem to study is usually difficult.

In real life, no one walks up to you and hands you an equation to be solved. Usually, it is a comment like “we need to make more money” or “we need to improve our efficiency.” Perhaps, we need to make better decisions or we need all our units that are not $100 \%$ efficient to become more efficient. We need to be precise in our formulation of the mathematics to actually describe the situation that we need to solve. In our example, we want to identify the spread of a contagious disease to determine how fast it will spread within our region. Perhaps, we will want to use the model to answer the following questions:

1. How long will it take until one thousand people get the disease?
2. What actions may be taken to slow or eradicate the disease?
Step 2: Make simplifying assumptions.
Giordano et al. (2014, pp. 62-65) described this well. Again, we suggest starting by brain storming the situation. Make a list of as many factors, or variables, as you can. Now, we realize that we usually cannot capture all these factors influencing a problem in our initial model. The task now is simplified by reducing the number of factors under consideration. We do this by making simplifying assumptions about the factors, such as holding certain factors as constants or ignoring some in the initial modeling phase. We might then examine to see if relationships exist between the remaining factors (or variables). Assuming simple relationships might reduce the complexity of the problem. Once you have a shorter list of variables, classify them as independent variables, dependent variables, or neither.

In our example, we assume we know the type of disease, how it is spread, the number of susceptible people within our region, and what type of medicine is needed to combat the disease. Perhaps, we assume that we know the size of population and the approximate number susceptible to getting the disease.

商业数学代考

1. 一千人得这种病需要多长时间？
2. 可以采取哪些行动来减缓或根除这种疾病？
第 2 步：做出简化假设。
佐丹奴等人。(2014, pp. 62-65) 很好地描述了这一点。同样，我们建议从头脑风暴开始。尽可能多地列出因素或变量。现在，我们意识到我们通常无法在初始模型中捕获影响问题的所有这些因素。现在通过减少所考虑因素的数量来简化任务。我们通过简化对因素的假设来做到这一点，例如将某些因素保持为常数或在初始建模阶段忽略一些因素。然后我们可能会检查其余因素（或变量）之间是否存在关系。假设简单的关系可能会降低问题的复杂性。一旦你有一个较短的变量列表，将它们分类为自变量、因变量或两者都不是。

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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|商业数学代写business mathematics代考|Overview and Process of Mathematical Modeling

Bender (2000, pp. 1-8) first introduced a process for modeling. He highlighted the following: formulate the model, outline the model, ask if it is useful, and test the model. Others have expanded this simple outlined process. Giordano et al. (2014, p. 64) presented a six-step process: identify the problem to be solved, make assumptions, solve the model, verify the model, implement the model, and maintain the model. Myer (2004, pp. 13-15) suggested some guidelines for modeling, including formulation, mathematical manipulation, and evaluation. Meerschaert (1999) developed a five-step process: ask the question, select the modeling approach, formulate the model, solve the model, and answer the question. Albright (2010) subscribed mostly to concepts and process described in previous editions of Giordano et al. (2014). Fox (2012, pp. 21-22) suggested an eight-step approach: understand the problem or question, make simplifying assumptions, define all variables, construct the model, solve and interpret the model, verify the model, consider the model’s strengths and weaknesses, and implement the model.
Most of these pioneers in modeling have suggested similar starts in understanding the problem or question to be answered and in making key assumptions to help enable the model to be built. We add the need for sensitivity analysis and model testing in this process to help ensure that we have a model that is performing correctly to answer the appropriate questions.

For example, student teams in the Mathematical Contest in Modeling were building models to determine the all-time best college sports coach. One team picked a coach who coached less than a year, went undefeated for the remaining part of the year, and won their bowl game. Thus, his season was a perfect season. Their algorithm picked this person as the all-time best coach. Sensitivity analysis and model testing could have shown the fallacy to their model.

Someplace between the defining of the variables and the assumptions, we begin to consider the model’s form and technique that might be used to solve the model. The list of techniques is boundless in mathematics, and we will not list them here. Suffice it to say that it might be good to initially decide among the forms: deterministic or stochastic for the model, linear or nonlinear for the relationship of the variables, and continuous or discrete.

We introduce the process of modeling and examine many different scenarios in which mathematical modeling can play a role.

The art of mathematical modcling is learned through expericnce of building and solving models. Modelers must be creative, innovative, inquisitive, and willing to try new techniques as well as being able to refine their models, if necessary. A major step in the process is passing the common sense test for use of the model.
In its basic form, modeling consists of three steps:

1. Make assumptions
2. Do some math
3. Derive and interpret conclusions
To that end, one cannot question the mathematics and its solution, but one can always question the assumptions used.

To gain insight, we will consider one framework that will enable the modeler to address the largest number of problems. The key is that there is something changing for which we want to know the effects and the results of the effects. The problem might involve any system under analysis. The realworld system can be very simplistic or very complicated. This requires both types of real-world systems to be modeled with the same logical stepwise process.

Consider modeling an investment. Our first inclination is to use the equations about compound interest rates that we used in high school or college algebra. The compound interest formula calculates the value of a compound interest investment after ” $n$ ” interest periods.
$$A=P(1-i)^{n}$$

where:
$A$ is the amount after $n$ interest periods
$P$ is the principal, the amount invested at the start $i$ is the interest rate applying to each period $n$ is the number of interest periods

商科代写|商业数学代写business mathematics代考|Overview and Process of Mathematical Modeling

Bender (2000, pp. 1-8) 首先介绍了一种建模过程。他强调了以下几点：制定模型、概述模型、询问它是否有用以及测试模型。其他人已经扩展了这个简单的概述过程。佐丹奴等人。(2014, p. 64) 提出了一个六步过程：识别要解决的问题、做出假设、解决模型、验证模型、实施模型和维护模型。Myer (2004, pp. 13-15) 提出了一些建模指南，包括公式化、数学操作和评估。Meerschaert (1999) 制定了一个五步流程：提出问题、选择建模方法、制定模型、解决模型和回答问题。Albright (2010) 主要赞同 Giordano 等人先前版本中描述的概念和过程。（2014）。福克斯（2012 年，第

1. 做出假设
2. 做一些数学
3. 推导和解释结论
为此，人们不能质疑数学及其解决方案，但人们总是可以质疑所使用的假设。

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。

商科代写|计量经济学代写Econometrics代考|ECON 2504

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics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。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

商科代写|计量经济学代写Econometrics代考|Some Examples

Each of the loss functions that we discuss in this subsection corresponds to a machine learning algorithm, as thoroughly explained in Bühlmann and Hothorn (2007), Sect. 3. We refer to this article for more properties of these losses and for issues regarding their practical implementation.

• A first canonical example, in the regression setting, is to let $\psi(x, y)=(y-x)^{2}$ (squared error loss), which is 2 -strongly convex in its first argument (Assumption $\mathbf{A}{\mathbf{2}}$ ) and satisfies Assumption $\mathbf{A}{\mathbf{1}}$ as soon as $\mathbb{E} Y^{2}<\infty$. It also satisfies $\mathbf{A}{\mathbf{3}}^{\prime}$, with $\partial{x} \psi(x, y)=2(x-y)$ and $L=2$.
• Another example in regression is the loss $\psi(x, y)=|y-x|$ (absolute error loss), which is convex but not strongly convex in its first argument. Whenever strong convexity of the loss is required, a possible strategy is to regularize the objective via an $L^{2}$-type penalty, and take
$$\psi(x, y)=|y-x|+\gamma x^{2},$$
where $\gamma$ is a positive parameter (possibly function of the sample size $n$ in the empirical setting). This loss is (2 ) -strongly convex in $x$ and satisfies $\mathbf{A}{\mathbf{1}}$ and $\mathbf{A}{\mathbf{2}}$ whenever $\mathbb{E}|Y|<\infty$, with $\xi(x, y)=\operatorname{sgn}(x-y)+2 \gamma x$ (with $\operatorname{sgn}(u)=2 \mathbb{1}{[u>0]}-1$ for $u \neq 0$ and $\operatorname{sgn}(0)=0$ ). On the other hand, the function $\psi(\cdot, y)$ is not differentiable at $y$, so that the smoothness Assumption $\mathbf{A}{3}^{\prime}$ is not satisfied. However,
\begin{aligned} \mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)=& \int\left(\operatorname{sgn}\left(x_{1}-y\right)-\operatorname{sgn}\left(x_{2}-y\right)\right) \mu_{Y \mid X}(\mathrm{~d} y)+2 \gamma\left(x_{1}-x_{2}\right) \ =& \mu_{Y \mid X}\left(\left(-\infty, x_{1}\right)\right)-\mu_{Y \mid X}\left(\left(-\infty, x_{2}\right)\right)+2 \gamma\left(x_{1}-x_{2}\right) \ &-\mu_{Y \mid X}\left(\left(x_{1}, \infty\right)\right)+\mu_{Y \mid X}\left(\left(x_{2}, \infty\right)\right) . \end{aligned}
Thus, if we assume for example that $\mu_{Y \mid X}$ has a density (with respect to the Lebesgue measure) bounded by $B$, then
$$\left|\mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)\right| \leq 2(B+\gamma)\left|x_{1}-x_{2}\right|,$$
and Assumption $\mathbf{A}{3}$ is therefore satisfied. Of course, in the empirical setting, assuming that $\mu{Y \mid X}$ has a density precludes the use of the empirical measure $\mu_{n}$ for $\mu_{X, Y}$. A safe and simple alternative is to consider a smoothed version $\tilde{\mu}{n}$ of $\mu{n}$ (based,
• for example, on a kernel estimate; see Devroye and Györfi 1985), and to minimize the functional
• $$• C_{n}(F)=\int|y-F(x)| \tilde{\mu}{n}(\mathrm{~d} x, \mathrm{~d} y)+\gamma \int F(x)^{2} \tilde{\mu}{n}(\mathrm{~d} x) •$$
• over the linear combinations of functions in $\mathscr{F}$.

商科代写|计量经济学代写Econometrics代考|Two Algorithms

Let $\operatorname{lin}(\mathscr{F})$ be the set of all linear combinations of functions in $\mathscr{F}$, our collection of base predictors in $L^{2}\left(\mu_{X}\right)$. So, each $F \in \operatorname{lin}(\mathscr{F})$ has the form $F=\sum_{j=1}^{J} \beta_{j} f_{j}$. where $\left(\beta_{1}, \ldots, \beta_{J}\right) \in \mathbb{R}^{J}$ and $f_{1}, \ldots, f_{J}$ are elements of $\mathscr{F}$. Finding the infimum of the functional $C$ over lin $(\mathscr{F})$ is a challenging infinite-dimensional optimization problem, which requires an algorithm. The core idea of the gradient boosting approach is to greedily locate the infimum by producing a combination of base predictors via a gradient-descent-type algorithm in $L^{2}\left(\mu_{X}\right)$. Focusing on the basics, this can be achieved by two related yet different strategies, which we examine in greater

mathematical details below. Algorithm 1 appears in Mason et al. (2000), whereas Algorithm 2 is essentially due to Friedman (2001).

It is implicitly assumed throughout this paragraph that Assumption $\mathbf{A}{\mathbf{1}}$ is satisfied. We recall that under this assumption, the convex functional $C$ is locally bounded and therefore continuous. Thus, in particular, $$\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F),$$ where $\varlimsup \overline{\operatorname{lin}(\mathscr{F})}$ is the closure of lin( $\mathscr{F})$ in $L^{2}\left(\mu{X}\right)$. Loosely speaking, looking for the infimum of $C$ over $\overline{\operatorname{lin}(\mathscr{F})}$ is the same as looking for the infimum of $C$ over all (finite) linear combinations of base functions in $\mathscr{F}$. We note in addition that if Assumption $\mathbf{A}{2}$ is satisfied, then there exists a unique function $\bar{F} \in \overline{\operatorname{lin}(\mathscr{F})}$ (which we call the boosting predictor) such that $$C(\bar{F})=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)$$
Algorithm 1. In this approach, we consider a class $\mathscr{F}$ of functions $f: \mathscr{X} \rightarrow \mathbb{R}$ such that $0 \in \mathscr{F}, f \in \mathscr{F} \Leftrightarrow-f \in \mathscr{F}$, and $|f|_{\mu_{X}}=1$ for $f \neq 0$. An example is the collection $\mathscr{F}$ of all $\pm$-binary trees in $\mathbb{R}^{d}$ using axis parallel cuts with $k$ terminal nodes (plus zero). Each nonzero $f \in \mathscr{F}$ takes the form $f=\sum_{j=1}^{k} \beta_{j} \mathbb{1}{A{j}}$, where $\left|\beta_{j}\right|=1$ and $A_{1}, \ldots, A_{k}$ is a tree-structured partition of $\mathrm{R}^{d}$ (Devroye et al. 1996, Chap. 20). The parameter $k$ is a measure of the tree complexity. For example, trees with $k=d+1$ are such that $\overline{\operatorname{lin}(\mathscr{F})}=L^{2}\left(\mu_{X}\right)$ (Breiman 2000). Thus, in this case,
$$\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)=\inf {F \in L^{2}\left(\mu_{X}\right)} C(F)$$

商科代写|计量经济学代写Econometrics代考|Algorithm 1

The convergence of this algorithm rests upon the choice of the step size sequence $\left(w_{t}\right){t}$, which needs to be carefully specified. We take $w{0}>0$ arbitrarily and set
$$w_{t+1}=\min \left(w_{t},-(2 L)^{-1} \mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X)\right), \quad t \geq 0,$$
where $L$ is the Lipschitz constant of Assumption $\mathbf{A}{\mathbf{3}}$. Clearly, the sequence $\left(w{t}\right){t}$ is nonincreasing. It is also nonnegative. To see this, just note that, by definition, $$f{t+1} \in \arg \max {f \in \mathscr{F}}-\mathbb{E} \xi\left(F{t}(X), Y\right) f(X),$$
and thus, since $0 \in \mathscr{F},-\mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X) \geq 0$. The main result of this section is encapsulated in the following theorem.

Theorem 1 Assume that Assumptions $\mathbf{A}{\mathbf{1}}$ and $\mathbf{A}{\mathbf{3}}$ are satisfied, and let $\left(F_{t}\right){t}$ be defined by Algorithm 1 with $\left(w{t}\right){t}$ as in (8). Then $$\lim {t \rightarrow \infty} C\left(F_{t}\right)=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F) .$$ Proof See Supplementary Material Document. Observe that Theorem 1 holds without Assumption $\mathbf{A}{2}$, i.e., there is no need here to assume that the function $\psi(x, y)$ is strongly convex in $x$. However, whenever Assumption $\mathbf{A}_{2}$ is satisfied, there exists as in (4) a unique boosting predictor $\bar{F} \in \overline{\operatorname{lin}(\mathscr{F})}$ such that $C(\bar{F})=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)$, and the theorem guarantees that $\lim {t \rightarrow \infty} C\left(F_{t}\right)=C(\bar{F})$

The proof of the theorem relies on the following lemma, which states that the sequence $\left(C\left(F_{t}\right)\right){t}$ is nonincreasing. Since $C(F)$ is nonnegative for all $F$, we concludê thât $C\left(F{t}\right) \downarrow \inf {k} C\left(F{k}\right)$ â $t \rightarrow \infty$. This is thé kêy argumént tó prové thé convergence of $C\left(F_{t}\right)$ toward inf $F \in \operatorname{lin}(\mathscr{F}) C(F)$.

商科代写|计量经济学代写Econometrics代考|Some Examples

• 第一个典型的例子，在回归设置中，是让 $\psi(x, y)=(y-x)^{2}$ (平方误差损失)，它的第一个参数是 2 – 强 凸 (假设 $\mathbf{A} 2$ ) 并满足假设 $\mathbf{A 1}$ 立刻 $\mathbb{E} Y^{2}<\infty$. 也满足 $\mathbf{A} \mathbf{3}^{\prime}$ ， 和 $\partial x \psi(x, y)=2(x-y)$ 和 $L=2$.
• 回归中的另一个例子是损失 $\psi(x, y)=|y-x|$ (绝对误差损失)，它是凸的，但在其第一个参数中不是强 凸的。每当需要损失的强凸性时，一种可能的策略是通过 $L^{2}$-型惩罚，并采取
$$\psi(x, y)=|y-x|+\gamma x^{2},$$
在哪里 $\gamma$ 是一个正参数 (可能是样本量的函数 $n$ 在经验设置中) 。这种损失是 (2) – 在 $x$ 并满足 $\mathbf{A} 1$ 和 $\mathbf{A} 2$ 每当 $\mathbb{E}|Y|<\infty$ ，和 $\xi(x, y)=\operatorname{sgn}(x-y)+2 \gamma x($ 和 $\operatorname{sgn}(u)=21[u>0]-1$ 为了 $u \neq 0$ 和 $\operatorname{sgn}(0)=0$ ) 。另一方面，函数 $\psi(\cdot, y)$ 不可微分 $y$, 使平滑假设 $\mathbf{A} 3^{\prime}$ 不满意。然而，
$$\mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)=\int\left(\operatorname{sgn}\left(x_{1}-y\right)-\operatorname{sgn}\left(x_{2}-y\right)\right) \mu_{Y \mid X}(\mathrm{~d} y)+2 \gamma\left(x_{1}-x_{2}\right)=$$
因此，如果我们假设例如 $\mu_{Y \mid X}$ 有一个密度（相对于 Lebesgue 测度) 为界 $B$ ，然后
$$\left|\mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)\right| \leq 2(B+\gamma)\left|x_{1}-x_{2}\right|,$$
和假设 $\mathbf{A} 3$ 因此感到满意。当然，在经验设置中，假设 $\mu Y \mid X$ 有一个密度排除了经验测量的使用 $\mu_{n}$ 为了 $\mu_{X, Y}$. 一个安全且简单的替代方案是考虑平滑版本 $\tilde{\mu} n$ 的 $\mu n$ (基于，
• 例如，在核估计上；参见 Devroye 和 Györfi 1985)，并最小化函数
• $\$ \$$• C_{n}(F)=|int |y F(x)| \backslash tilde {\backslash m u}{n}(\backslash \operatorname{mathrm}{\sim d} x, \mathrm {\sim d} y)+\backslash gamma \backslash int F(x)^{\wedge}{2} \backslash t \mathrm{~ i l d e {} (数学 {\sim d} x) • \\ • 在函数的线性组合上 \mathscr{F}. 商科代写|计量经济学代写Econometrics代考|Two Algorithms 让 \operatorname{lin}(\mathscr{F}) 是函数的所有线性组合的集合 \mathscr{F} ，我们收集的基础预测变量在 L^{2}\left(\mu_{X}\right). 所以，每个 F \in \operatorname{lin}(\mathscr{F}) 有形 式 F=\sum_{j=1}^{J} \beta_{j} f_{j}. 在哪里 \left(\beta_{1}, \ldots, \beta_{J}\right) \in \mathbb{R}^{J} 和 f_{1}, \ldots, f_{J} 是元素 \mathscr{F}. 寻找泛函的下确界 C 过林 (\mathscr{F}) 是一个具 有挑战性的无限维优化问题，需要一个算法。梯度提升方法的核心思想是通过梯度下降型算法生成基本预测变量的 组合来贪婪地定位下确界。 L^{2}\left(\mu_{X}\right). 专注于基础，这可以通过两种相关但不同的策略来实现，我们将在更大的 下面的数学细节。算法 1 出现在 Mason et al. (2000 年)，而算法 2 主要归功于 Friedman（2001 年)。 在本段中隐含地假设假设 \mathbf{A 1} 很满意。我们记得在这个假设下，凸泛函 C 是局部有界的，因此是连续的。因此，特 别是，$$
\inf F \in \operatorname{lin}(\mathscr{F}) C(F)=\inf F \in \operatorname{lin}(\mathscr{F}) C(F),
$$在哪里 \overline{\lim } \overline{\operatorname{lin}(\mathscr{F})} 是 \operatorname{lin}(\mathscr{F}) 在 L^{2}(\mu X). 松散地说，寻找下确界 C 超过 \overline{\operatorname{lin}(\mathscr{F}) \text { 和寻找下确界一样 } C \text { 基函数的所 } 有 (有限) 线性组合 \mathscr{F}. 我们还注意到，如果假设 \mathbf{A} 2 满足，则存在唯一函数 \bar{F} \in \overline{\operatorname{lin}(\mathscr{F})} (我们称之为提升预测 器) 使得$$
C(\bar{F})=\inf F \in \operatorname{lin}(\mathscr{F}) C(F)
$$算法 1. 在这种方法中，我们考虑一个类 \mathscr{F} 功能 f: \mathscr{X} \rightarrow \mathbb{R} 这样 0 \in \mathscr{F}, f \in \mathscr{F} \Leftrightarrow-f \in \mathscr{F} ，和 |f|{\mu{X}}=1 为了 f \neq 0. 一个例子是集合 \mathscr{F} 其中 \pm – 二叉树 \mathbb{R}^{d} 使用轴平行切割 k 终端节点 (加零) 。每个非零 f \in \mathscr{F} 采取形式 f=\sum_{j=1}^{k} \beta_{j} 1 A j ，在哪里 \left|\beta_{j}\right|=1 和 A_{1}, \ldots, A_{k} 是一个树形结构的分区 \mathrm{R}^{d} (Devroye 等人，1996 年，第 20 章) 。参数 k 是树复杂度的度量。例如，树与 k=d+1 是这样的 \overline{l i n}(\mathscr{F})=L^{2}\left(\mu_{X}\right) (布雷曼 2000) 。因 此，在这种情况下，$$
\inf F \in \operatorname{lin}(\mathscr{F}) C(F)=\inf F \in L^{2}\left(\mu_{X}\right) C(F)
$$商科代写|计量经济学代写Econometrics代考|Algorithm 1 该算法的收敛取决于步长序列的选择 \left(w_{t}\right) t, 这需要仔细指定。我们采取 w 0>0 任意设置$$
w_{t+1}=\min \left(w_{t},-(2 L)^{-1} \mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X)\right), \quad t \geq 0,
$$在哪里 L 是假设的 Lipschitz 常数 \mathbf{A 3}. 显然，序列 (w t) t 是非增加的。它也是非负的。要看到这一点，请注意，根 据定义，$$
f t+1 \in \arg \max f \in \mathscr{F}-\mathbb{E} \xi(F t(X), Y) f(X),
$$因此，由于 0 \in \mathscr{F},-\mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X) \geq 0. 本节的主要结果封装在以下定理中。 定理 1 假设假设 \mathbf{A} 1 和 \mathbf{A} 3 满意，让 \left(F_{t}\right) t 由算法 1 定义 (w t) t 如（8)。然后$$
\lim t \rightarrow \infty C\left(F_{t}\right)=\inf F \in \operatorname{lin}(\mathscr{F}) C(F) .


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。