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## 统计代写|概率模型代写Statistical Model代考|Third assumption

If we also assume that the observations are conditionally independent, then the least-squares estimates have the smallest variance among all unbiased linear estimates. Since the estimates were computed by minimizing the residual variation, this outcome should not be particularly surprising. The assumption of conditional independence is harder to check definitively. Checking this assumption appropriately will usually require the use of some knowledge about the design of the data collection. Generally the analyst will use information about the design to guide the choice of the types of dependence to check. For example, if groups of the observations have similar origin, clustering, then it may be worth checking for intra-group correlation, and if the data have a time stamp then checking for autocorrelation is an important consideration.

## 统计代写|概率模型代写Statistical Model代考|Fourth condition

This point is mentioned because it is relevant here, although it resides more naturally with Section 3.3. It is not really an assumption about the conditional distribution of the response variable, as such. Least-squares estimates can be expressed as sums of conditionally independent random variables, so the estimates are subject to the Central Limit Theorem. The interested reader can learn more from Huber (1981, Theorem 2.3, Chapter 7), Demidenko (2004, $\S 13.1 .1)$, and DasGupta (2008, Theorem $5.3$ and Example 5.1). Consequently, asymptotically, the estimates are normally distributed. This observation can be used to justify an assumption of normality for the parameter estimates, which can in turn be used to construct interval estimates and hypothesis tests. However, the assumption of conditionally normal errors, as in Section 3.3, is more commonly used. We discuss this point further in Section 3.4.8.4.

## 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。

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## 统计代写|概率模型代写Statistical Model代考|Ordinary Regression

Linear regression and its generalization, the linear model, are in very common use in statistics. For example, Jennrich (1984) wrote, “I have long been a proponent of the following unified field theory for statistics: Almost all of statistics is linear regression, and most of what is left over is non-linear regression.” This is hardly surprising when we consider that linear regression focuses on estimating the first derivative of relationships between variables, that is, rates of change. The most common uses to which the linear regression model is put are

1. to enable prediction of a random variable at specific combinations of other variables;
2. to estimate the effect of one or more variables upon a random variable; and
3. to nominate a subset of variables that is most influential upon a random variable.
Linear regression provides a statistical answer to the question of how a target variable (usually called the response or dependent variable) is related to one or more other variables (usually called the predictor or independent variables). Linear regression both estimates and assesses the strength of the statistical patterns of covariation. However, it makes no comment on the causal strength of any pattern that it identifies.

The algebraic expression of the linear regression model for one predictor variable and one response variable is
$$y_i=\beta_0+\beta_1 \times x_i+\epsilon_i$$
where $y_i$ is the value of the response variable for the $i$-th observation, $x_i$ and $\epsilon_i$ are similarly the predictor variable and the error respectively, and $\beta_0$ and $\beta_1$ are the unknown intercept and slope of the relationship between the random variables $x$ and $y$.

## 统计代写|概率模型代写Statistical Model代考|Least-Squares Regression

The challenge of determining estimates for the parameters, conditional on data, can be framed as an optimization problem. For least-squares regression, we are interested in finding the values of the parameters that minimize the sum of the squared residuals, where the residuals are defined as the differences between the observed values of $y$ and the predicted values of $y$, called $\hat{y}$.
Exact solutions are available for least-squares linear regression, but our ultimate goal is to develop models for which no exact solutions exist. Therefore we treat least-squares linear regression in this manner as an introduction.
$$\beta_0^{\min }, \beta_1 \sum_{i=1}^n\left(y_i-\left(\beta_0+\beta_1 x_i\right)\right)^2$$
For example, consider the following observations, for which least-squares optimization is decidedly unnecessary.
\begin{aligned} &>y<-c(3,5,7) \ &>x<-c(1,2,3) \end{aligned} We can write the objective function as a function in $\mathrm{R}$, and use the powerful optim function to minimize the objective function across its first argument, which may be of any length. So, the least-squares objective function for obtaining estimates of $\beta_0$ and $\beta_1$ can be written in $\mathrm{R}$ as $>$ least.squares <- function $(p, x, y){$
$+\operatorname{sum}((y-(p[1]+p[2] * x)) \sim 2)$
$+}$
where $\mathrm{x}$ is the predictor variable, $\mathrm{y}$ is the response variable, and $\mathrm{p}$ is the vector of parameters.

## 统计代写|概率模型代写Statistical Model代考|Ordinary Regression

$$y_i=\beta_0+\beta_1 \times x_i+\epsilon_i$$

## 统计代写|概率模型代写Statistical Model代考|Least-Squares Regression

$$\beta_0^{\min }, \beta_1 \sum_{i=1}^n\left(y_i-\left(\beta_0+\beta_1 x_i\right)\right)^2$$

$$y<-c(3,5,7) \quad>x<-c(1,2,3)$$ 我们可以把目标函数写成一个函数 $R$, 并使用强大的 optim 函数来最小化其第一个参数的目标函数，该参数可以是任意长度。因 此，用于获得估计的最小二乘目标函数 $\beta_0$ 和 $\beta_1$ 可以写成R作为 $>$ 最小二乘 $<-$ 函数
$(p, x, y) \$ \$+\operatorname{sum}((y-(p[1]+p[2] * x)) \sim 2) \$ \$+$

## 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。

Posted on Categories:Statistical Model, 数据科学代写, 概率模型, 统计代写, 统计代考

## avatest™帮您通过考试

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## 统计代写|概率模型代写Statistical Model代考|Process

Recall that the probabilities that are derived from a PDF are described by parameters. When we are modelling with data, we want to estimate the pa-rameters of the model using the data. The parameters of a probability function are usually not directly estimated in statistical modelling. Instead, the conditioning of the PF is reversed. When the relationship of observations to parameters are reversed for a given probability function, statisticians refer to the function as a likelihood function. For a given probability distribution, we may write $f(y \mid \theta)$ where $y$ represents the data and $\theta$ is the distribution parameter that produces $y$. Then the corresponding likelihood function is $L(\theta \mid y)$. The functional form is identical; all that changes is the conditioning. The probability refers to the probability of data conditional on parameters, whereas the likelihood refers to the likelihood of parameters conditional on data.

When models are estimated using maximum likelihood, the likelihood is transformed by the natural logarithm so that the contributions from each unit of the dataset are summed (under the assumption of conditional independence of the observations of the population), instead of being multiplied. This is because summing across values is numerically more stable than is multiplying across values. We will reserve $L(\theta \mid y)$ to refer to the log-likelihood of the parameters conditional on the data.

For an example we consider a Poisson model. The probability distribution for a single observation is
$$f_{Y=y}(y \mid \lambda)=\frac{\lambda^y e^{-\lambda}}{y !}$$
where $y$ is the response variable and $\lambda$ is the mean or location parameter. The data are determined by the mean parameter via the PDF. A product sign would be placed in front of the probability function for an independent and identically distributed (iid) sample of observations.

## 统计代写|概率模型代写Statistical Model代考|Estimation

We now demonstrate maximum likelihood estimation of the single parameter of Watson’s distribution, using $\mathrm{R}$ code. Recall from the previous chapter that the PDF is
$$f(x ; \theta)=\frac{1+\theta}{\theta\left(1+\frac{x}{\theta}\right)^2} \quad 00$$
This equation translates to the following log-likelihood.
$$\mathcal{L}(\theta ; x)=\log (1+\theta)-\log (\theta)-2 \times \log \left(1+\frac{x}{\theta}\right) \quad 00$$
In $\mathrm{R}$, for a vector of data $\mathrm{x}$, the function is as follows.
$>$ jll.watson <- function(theta, $x){$
$+\operatorname{sum}(\log (1+$ theta $)-\log ($ theta $)-2 * \log (1+x /$ theta $))$
$+3$
We can maximize this function across $\theta$ a number of ways. We will use the optim function here, and we write a wrapper function for it to simplify our future usage. Our wrapper function is

## 统计代写概率模型代写Statistical Model代考|Process

$$f_{Y=y}(y \mid \lambda)=\frac{\lambda^y e^{-\lambda}}{y !}$$

## 统计代写|概率模型代写Statistical Model代考|Estimation

$$f(x ; \theta)=\frac{1+\theta}{\theta\left(1+\frac{x}{\theta}\right)^2} \quad 00$$

$$\mathcal{L}(\theta ; x)=\log (1+\theta)-\log (\theta)-2 \times \log \left(1+\frac{x}{\theta}\right) \quad 00$$

$>$ jll.watson <- 函数 $(\theta, \$ \mathrm{x}){+\backslash$操作员名称${$sum$}(\backslash \log (1+$theta$)-\backslash$日志 (theta)$-2 * \backslash \log (1+\mathrm{x} /$theta$))+3$Wecanmaximizethis functionacross \theta\$ 多种方式。我们将在这里使用 optim 函数，并为它编写一个包装函数以简 化㑘们末来的使用。我们的包装函数是

## 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。