### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|MAST90085

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• 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

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Transformations

Suppose that $X$ has pdf $f_{X}(x)$. What is the pdf of $Y=3 X$ ? Or if $X=$ $\left(X_{1}, X_{2}, X_{3}\right)^{\top}$, what is the pdf of
$$Y=\left(\begin{array}{c} 3 X_{1} \ X_{1}-4 X_{2} \ X_{3} \end{array}\right) ?$$
This is a special case of asking for the pdf of $Y$ when
$$X=u(Y)$$
for a one-to-one transformation $u: \mathbb{R}^{p} \rightarrow \mathbb{R}^{p}$. Define the Jacobian of $u$ as
$$\mathcal{J}=\left(\frac{\partial x_{i}}{\partial y_{j}}\right)=\left(\frac{\partial u_{i}(y)}{\partial y_{j}}\right)$$
and let $\operatorname{abs}(|\mathcal{J}|)$ be the absolute value of the determinant of this Jacobian. The pdf of $Y$ is given by
$$f_{Y}(y)=\operatorname{abs}(|\mathcal{J}|) \cdot f_{X}{u(y)}$$
Using this we can answer the introductory questions, namely
$$\left(x_{1}, \ldots, x_{p}\right)^{\top}=u\left(y_{1}, \ldots, y_{p}\right)=\frac{1}{3}\left(y_{1}, \ldots, y_{p}\right)^{\top}$$
with
$$\mathcal{J}=\left(\begin{array}{ccc} \frac{1}{3} & & 0 \ & \ddots & \ 0 & & \frac{1}{3} \end{array}\right)$$
and hence $\operatorname{abs}(|\mathcal{J}|)=\left(\frac{1}{3}\right)^{p}$. So the pdf of $Y$ is $\frac{1}{3^{p}} f_{X}\left(\frac{y}{3}\right)$.
This introductory example is a special case of
$$Y=\mathcal{A} X+b \text {, where } \mathcal{A} \text { is nonsingular. }$$
The inverse transformation is
$$X=\mathcal{A}^{-1}(Y-b)$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Multinormal Distribution

The multinormal distribution with mean $\mu$ and covariance $\Sigma>0$ has the density
$$f(x)=|2 \pi \Sigma|^{-1 / 2} \exp \left{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1}(x-\mu)\right}$$
We write $X \sim N_{p}(\mu, \Sigma)$.
How is this multinormal distribution with mean $\mu$ and covariance $\Sigma$ related to the multivariate standard normal $N_{p}\left(0, \mathcal{I}_{p}\right)$ ? Through a linear transformation using the results of Sect. $4.3$, as shown in the next theorem.

Theorem 4.5 Let $X \sim N_{p}(\mu, \Sigma)$ and $Y=\Sigma^{-1 / 2}(X-\mu)$ (Mahalanobis transfor mation). Then
$$Y \sim N_{p}\left(0, \mathcal{I}{p}\right),$$ i.e. the elements $Y{j} \in \mathbb{R}$ are independent, one-dimensional $N(0,1)$ variables.
Proof Note that $(X-\mu)^{\top} \Sigma^{-1}(X-\mu)=Y^{\top} Y$. Application of (4.45) gives $\mathcal{J}=$ $\Sigma^{1 / 2}$, hence
$$f_{Y}(y)=(2 \pi)^{-p / 2} \exp \left(-\frac{1}{2} y^{\top} y\right)$$
which is by $(4.47)$ the pdf of a $N_{p}\left(0, \mathcal{I}_{p}\right)$.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Sampling Distributions and Limit Theorems

In multivariate statistics, we observe the values of a multivariate random variable $X$ and obtain a sample $\left{x_{i}\right}_{i=1}^{n}$, as described in Chap. 3. Under random sampling, these observations are considered to be realisations of a sequence of i.i.d. random variables $X_{1}, \ldots, X_{n}$, where each $X_{i}$ is a $p$-variate random variable which replicates the parent or population random variable $X$. Some notational confusion is hard to avoid: $X_{i}$ is not the $i$ th component of $X$, but rather the $i$ th replicate of the $p$-variate random variable $X$ which provides the $i$ th observation $x_{i}$ of our sample.

For a given random sample $X_{1}, \ldots, X_{n}$, the idea of statistical inference is to analyse the properties of the population variable $X$. This is typically done by analysing some characteristic $\theta$ of its distribution, like the mean, covariance matrix, etc. Statistical inference in a multivariate setup is considered in more detail in Chaps. 6 and $7 .$

Inference can often be performed using some observable function of the sample $X_{1}, \ldots, X_{n}$, i.e. a statistics. Examples of such statistics were given in Chap. 3: the sample mean $\bar{x}$, the sample covariance matrix $\mathcal{S}$. To get an idea of the relationship between a statistics and the corresponding population characteristic, one has to derive the sampling distribution of the statistic. The next example gives some insight into the relation of $(\bar{x}, S)$ to $(\mu, \Sigma)$.

Example $4.15$ Consider an iid sample of $n$ random vectors $X_{i} \in \mathbb{R}^{p}$ where $\mathrm{E}\left(X_{i}\right)=\mu$ and $\operatorname{Var}\left(X_{i}\right)=\Sigma$. The sample mean $\bar{x}$ and the covariance matrix $\mathcal{S}$ have already been defined in Sect. 3.3. It is easy to prove the following results:
\begin{aligned} &\mathrm{E}(\bar{x})=n^{-1} \sum_{i=1}^{n} \mathrm{E}\left(X_{i}\right)=\mu \ &\operatorname{Var}(\bar{x})=n^{-2} \sum_{i=1}^{n} \operatorname{Var}\left(X_{i}\right)=n^{-1} \Sigma=\mathrm{E}\left(\bar{x} \bar{x}^{\top}\right)-\mu \mu^{\top} \end{aligned}

\begin{aligned} \mathrm{E}(\mathcal{S}) &=n^{-1} \mathrm{E}\left{\sum_{i=1}^{n}\left(X_{i}-\bar{x}\right)\left(X_{i}-\bar{x}\right)^{\top}\right} \ &=n^{-1} \mathrm{E}\left{\sum_{i=1}^{n} X_{i} X_{i}^{\top}-n \bar{x} \bar{x}^{\top}\right} \ &=n^{-1}\left{n\left(\Sigma+\mu \mu^{\top}\right)-n\left(n^{-1} \Sigma+\mu \mu^{\top}\right)\right} \ &=\frac{n-1}{n} \Sigma . \end{aligned}
This shows in particular that $\mathcal{S}$ is a biased estimator of $\Sigma$. By contrast, $\mathcal{S}_{u t}=\frac{n}{n-1} \mathcal{S}$ is an unbiased estimator of $\Sigma$.

Statistical inference often requires more than just the mean and/or the variance of a statistic. We need the sampling distribution of the statistics to derive confidence intervals or to define rejection regions in hypothesis testing for a given significance level. Theorem $4.9$ gives the distribution of the sample mean for a multinormal population.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Transformations

X=在(是)

Ĵ=(∂X一世∂是j)=(∂在一世(是)∂是j)

F是(是)=腹肌⁡(|Ĵ|)⋅FX在(是)

(X1,…,Xp)⊤=在(是1,…,是p)=13(是1,…,是p)⊤

Ĵ=(130 ⋱ 013)

X=一个−1(是−b)

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Multinormal Distribution

f(x)=|2 \pi \Sigma|^{-1 / 2} \exp \left{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1 }(x-\mu)\right}f(x)=|2 \pi \Sigma|^{-1 / 2} \exp \left{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1 }(x-\mu)\right}

F是(是)=(2圆周率)−p/2经验⁡(−12是⊤是)

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Sampling Distributions and Limit Theorems

\begin{对齐} \mathrm{E}(\mathcal{S}) &=n^{-1} \mathrm{E}\left{\sum_{i=1}^{n}\left(X_{i }-\bar{x}\right)\left(X_{i}-\bar{x}\right)^{\top}\right} \ &=n^{-1} \mathrm{E}\left {\sum_{i=1}^{n} X_{i} X_{i}^{\top}-n \bar{x} \bar{x}^{\top}\right} \ &=n^ {-1}\left{n\left(\Sigma+\mu \mu^{\top}\right)-n\left(n^{-1} \Sigma+\mu \mu^{\top}\right) \right} \ &=\frac{n-1}{n} \Sigma 。\end{对齐}\begin{对齐} \mathrm{E}(\mathcal{S}) &=n^{-1} \mathrm{E}\left{\sum_{i=1}^{n}\left(X_{i }-\bar{x}\right)\left(X_{i}-\bar{x}\right)^{\top}\right} \ &=n^{-1} \mathrm{E}\left {\sum_{i=1}^{n} X_{i} X_{i}^{\top}-n \bar{x} \bar{x}^{\top}\right} \ &=n^ {-1}\left{n\left(\Sigma+\mu \mu^{\top}\right)-n\left(n^{-1} \Sigma+\mu \mu^{\top}\right) \right} \ &=\frac{n-1}{n} \Sigma 。\end{对齐}

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