### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Inference for a Multivariate Normal

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

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

In previous chapters, we have mostly focused on multivariate exploratory analyses, without statistical inference. That is, our focus is mostly on finding important descriptive features of data, such as data dimensions and data classifications, without extending the results or conclusions to the whole population. Distributional assumptions for the data are often not required for such exploratory analysis. Exploratory data analysis is an important component of statistical analysis. However, exploratory analysis is not sufficient. Often we wish to extend the conclusions from exploratory analysis to the whole population. That is, we also wish to make statistical inference, such as hypothesis testing or confidence intervals.

In this chapter, we consider statistical inference for multivariate continuous data, i.e., we try to extend the results from the data to the whole population. In order to make inference, the following assumptions are often required: (i) the sample is a random and representative subset of the population, e.g., an i.i.d. (independent and identically distributed) sample from the population, and (ii) the population is assumed to follow a parametric distribution, such as a multivariate normal distribution. In this chapter, we focus on the most important distribution for multivariate continuous data, i.e., the multivariate normal distribution. A multivariate normal distribution, denoted by $N(\mu, \Sigma)$, is completely determined by its mean vector $\mu=\left(\mu_{1}, \mu_{2}, \cdots, \mu_{p}\right)^{\mathrm{T}}$ and its covariance matrix $\Sigma=\left(\sigma_{i j}\right)_{p \times p}$.

Inference for a multivariate normal distribution is often the main focus of many classic multivariate analysis books. Many nice features of the multivariate normal distribution allow us to do theoretical arguments and derive many elegant results. In practice, a multivariate normal distributional assumption may be reasonable for continuous data in many cases (perhaps after some transformations of the original data), especially when the sample size is large so the central limit theorems can be used to justify the normality assumption for the sample means. In some other cases in practice, such as small sample sizes or skewed data or discrete data, the multivariate

normal assumption may be too strong or unreasonable. In real data analysis, when we use a method or model which require normality assumption, we should check to see if this assumption is indeed reasonable.

In the following sections, we consider inference for both the mean vector and the covariance matrix of a multivariate normal distribution. Since theoretical derivations of these results are available in many classic books and our focus is to use these methods in practice, we skip the theoretical derivations and focus on explaining the methods and their applicability in data analysis. Readers interested in mathematical derivations are referred to many classic multivariate analysis books (e.g., Johnson and Wichern 2007).

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Inference for Multivariate Means

We first review the well-known $t$-test for inference of a univariate mean. In univariate analysis, the $t$-test is widely used to test the mean parameter assumed for a continuous variable. Suppose that $\left{x_{1}, x_{2}, \cdots, x_{n}\right}$ is a random sample from a univariate normal population $N\left(\mu, \sigma^{2}\right)$. Consider a two-sided test for the mean parameter $\mu$ :
$$H_{0}: \mu=\mu_{0} \quad \text { versus } \quad H_{1}: \mu \neq \mu_{0}$$
where $\mu_{0}$ is known. Let the sample mean and the sample standard deviation be $\bar{x}$ and $s$ respectively. The $t$ test statistic is given by
$$t=\frac{\sqrt{n}\left(\bar{x}-\mu_{0}\right)}{s}$$
which is in fact a standardize version of the sample mean $\bar{x}$ under $H_{0}$. Under the null hypothesis $H_{0}$, the test statistic $t$ follows a $t$-distribution with $n-1$ degrees of freedom. An alternative and equivalent test statistic is given by
$$t^{2}=\frac{n\left(\bar{x}-\mu_{0}\right)^{2}}{s^{2}},$$
which follows a $F(1, n-1)$-distribution. These tests are relatively robust against small to moderate departure from the assumed normality of the population, especially when the sample size is large. In fact, when the sample size is large, the sample mean $\bar{x}$ will approximately follow a normal distribution for any population distribution, based on the central limit theorem. In other words, $t$-tests can be used as long as the sample size is large, even if the population is not normally distributed.

The above univariate $t$-test can be extended to the multivariate case. For multivariate data, a key consideration is to incorporate the correlation or covariance between the variables. The most well-known extension is the so-called Hotelling’s $T$ test, which is described as follows. Let $\mathbf{x}{1}, \cdots, \mathbf{x}{n}$ be a random sample from the

multivariate normal population $N_{p}(\mu, \Sigma)$, where $\mathbf{x}{i}=\left(x{i 1}, \cdots, x_{i p}\right)^{\mathrm{T}}$ and both $\boldsymbol{\mu}$ and $\Sigma$ are unknown. Let
$$\overline{\mathrm{x}}=\frac{\sum_{i=1}^{n} \mathbf{x}{i}}{n}, \quad \boldsymbol{S}=\frac{1}{n-1} \sum{i=1}^{n}\left(\mathbf{x}{i}-\overline{\mathrm{x}}\right)\left(\mathbf{x}{i}-\overline{\mathrm{x}}\right)^{\mathrm{T}}$$
be the sample mean vector and sample covariance matrix respectively. Suppose that we wish to test the following two-sided multivariate hypotheses
$$H_{0}: \boldsymbol{\mu}=\boldsymbol{\mu}{0} \quad \text { versus } \quad H{1}: \boldsymbol{\mu} \neq \boldsymbol{\mu}{0},$$ where $\boldsymbol{\mu}{0}$ is a known vector. The Hotelling’s $T^{2}$ test statistic is given by
$$T^{2}=n\left(\overline{\mathrm{x}}-\mu_{0}\right)^{\mathrm{T}} S^{-1}\left(\overline{\mathrm{x}}-\mu_{0}\right) .$$
Under $H_{0}$, we have
$$T^{* 2}=\frac{n-p}{p(n-1)} T^{2} \sim F(p, n-p) .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Inference for Covariance Matrices

For a multivariate normal population $N_{p}(\boldsymbol{\mu}, \Sigma)$, inference for the covariance matrix $\Sigma$ can also be performed. However, the computation associated with the test can be tedious since closed-form null distributions of test statistics are often unavailable, so computer software is needed for computation.

In practice, it is common to test the equality of two covariance matrices, For example, when testing two multivariate mean vectors, it is assumed that the two unknown covariance matrices are equal (as noted in the previous section). This assumption can be tested. Specifically, we can test the equality of two population covariance matrices
$$H_{0}: \Sigma_{1}=\Sigma_{2} \quad \text { versus } \quad \Sigma_{1} \neq \Sigma_{2} \text {. }$$
Let $\left{\mathbf{x}{1}, \cdots, \mathbf{x}{n_{1}}\right}$ be a random sample from population $N_{p}\left(\boldsymbol{\mu}{1}, \Sigma{1}\right)$, and let $\left{\mathbf{y}{1}, \cdots\right.$, $\left.\mathbf{y}{n_{2}}\right}$ be an independent random sample from population $N_{p}\left(\mu_{2}, \Sigma_{2}\right)$. The test statistic is given by
$$\Lambda=\frac{\left|\hat{\Sigma}{1}\right|^{\left(n{1}-1\right) / 2}\left|\hat{\Sigma}{2}\right|^{\left(n{2}-1\right) / 2}}{|\hat{\Sigma}|(n-2) / 2}$$

where $\hat{\Sigma}{1}$ and $\hat{\Sigma}{2}$ are sample covariance matrices of $\Sigma_{1}$ and $\Sigma_{2}$ respectively,
$$\hat{\Sigma}=\left(\left(n_{1}-1\right) \hat{\Sigma}{1}+\left(n{2}-1\right) \hat{\Sigma}{2}\right) /(n-2)$$ is the pooled estimate of the covariance matrix, and $n=n{1}+n_{2}$ is the total sample size. The null distribution of $\Lambda$ is complicated, but computer software can be used to obtain p-values of the test.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Inference for Multivariate Means

H0:μ=μ0 相对 H1:μ≠μ0

X¯=∑一世=1nX一世n,小号=1n−1∑一世=1n(X一世−X¯)(X一世−X¯)吨

H0:μ=μ0 相对 H1:μ≠μ0,在哪里μ0是一个已知向量。霍特林的吨2检验统计量由下式给出

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Inference for Covariance Matrices

H0:Σ1=Σ2 相对 Σ1≠Σ2.

Λ=|Σ^1|(n1−1)/2|Σ^2|(n2−1)/2|Σ^|(n−2)/2

Σ^=((n1−1)Σ^1+(n2−1)Σ^2)/(n−2)是协方差矩阵的合并估计，并且n=n1+n2是总样本量。的零分布Λ很复杂，但是可以使用计算机软件来获得检验的 p 值。

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## MATLAB代写

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