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# 统计代写|概率模型代写Statistical Model代考|STAT202 Maximum Likelihood Estimation

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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。