## 统计代写|线性回归代写linear regression代考|Multivariate Models

The multivariate location and dispersion model is a special case of the multivariate linear regression model when the design matrix is equal to the vector of ones: $\boldsymbol{X}=\mathbf{1}$. See Chapter 12. (Similarly, the location model is a special case of the multiple linear regression model. See Section 2.9.1.) The multivariate normal and elliptically contoured distributions are important parametric models for the multivariate location and dispersion model. The multivariate normal distribution is useful in the large sample theory of the linear model, covered in Chapter 11, while elliptically contoured distributions are useful for multivariate linear regression. Section 3.4.1 used prediction regions for iid multivariate data to bootstrap hypothesis tests.

Definition 10.1. An important multivariate location and dispersion model is a joint distribution with joint pdf
$$f(\boldsymbol{z} \mid \boldsymbol{\mu}, \boldsymbol{\Sigma})$$
for a $p \times 1$ random vector $\boldsymbol{x}$ that is completely specified by a $p \times 1$ population location vector $\boldsymbol{\mu}$ and a $p \times p$ symmetric positive definite population dispersion matrix $\boldsymbol{\Sigma}$. Thus $P(\boldsymbol{x} \in A)=\int_A f(\boldsymbol{z}) d \boldsymbol{z}$ for suitable sets $A$.

The multivariate location and dispersion model is in many ways similar to the multiple linear regression model. The data are iid vectors from some distribution such as the multivariate normal (MVN) distribution. The location parameter $\boldsymbol{\mu}$ of interest may be the mean or the center of symmetry of an elliptically contoured distribution. Hyperellipsoids will be estimated instead of hyperplanes, and Mahalanohis distances will be used instead of ahsolute residuals to determine if an observation is a potential outlier.

Assume that $\boldsymbol{X}_1, \ldots, \boldsymbol{X}_n$ are $n$ iid $p \times 1$ random vectors and that the joint pdf of $\boldsymbol{X}_1$ is $f(\boldsymbol{z} \mid \boldsymbol{\mu}, \boldsymbol{\Sigma})$.

## 统计代写|线性回归代写linear regression代考|The Multivariate Normal Distribution

Definition 10.2: Rao $(1965$, p. 437$)$. A $p \times 1$ random vector $\boldsymbol{X}$ has a $p$-dimensional multivariate normal distribution $N_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ iff $\boldsymbol{t}^T \boldsymbol{X}$ has a univariate normal distribution for any $p \times 1$ vector $t$.
If $\boldsymbol{\Sigma}$ is positive definite, then $\boldsymbol{X}$ has a pdf
$$f(\boldsymbol{z})=\frac{1}{(2 \pi)^{p / 2}|\boldsymbol{\Sigma}|^{1 / 2}} e^{-(1 / 2)\left(\boldsymbol{z}{-} \boldsymbol{\mu}\right)^T \boldsymbol{\Sigma}^{-1}\left(\boldsymbol{z}{-} \boldsymbol{\mu}\right)}$$
where $|\Sigma|^{1 / 2}$ is the square root of the determinant of $\boldsymbol{\Sigma}$. Note that if $p=1$, then the quadratic form in the exponent is $(z-\mu)\left(\sigma^2\right)^{-1}(z-\mu)$ and $X$ has the univariate $N\left(\mu, \sigma^2\right)$ pdf. If $\Sigma$ is positive semidefinite but not positive definite, then $\boldsymbol{X}$ has a degenerate distribution. For example, the univariate $N\left(0,0^2\right)$ distribution is degenerate (the point mass at 0 ).

Definition 10.3. The population mean of a random $p \times 1$ vector $\boldsymbol{X}=$ $\left(X_1, \ldots, X_p\right)^T$ is
$$E(\boldsymbol{X})=\left(E\left(X_1\right), \ldots, E\left(X_p\right)\right)^T$$
and the $p \times p$ population covariance matrix
$$\operatorname{Cov}(\boldsymbol{X})=E(\boldsymbol{X}-E(\boldsymbol{X}))(\boldsymbol{X}-E(\boldsymbol{X}))^T=\left(\sigma_{i j}\right) .$$
That is, the $i j$ entry of $\operatorname{Cov}(\boldsymbol{X})$ is $\operatorname{Cov}\left(X_i, X_j\right)=\sigma_{i j}$.
The covariance matrix is also called the variance-covariance matrix and variance matrix. Sometimes the notation $\operatorname{Var}(\boldsymbol{X})$ is used. Note that $\operatorname{Cov}(\boldsymbol{X})$ is a symmetric positive semidefinite matrix. If $\boldsymbol{X}$ and $\boldsymbol{Y}$ are $p \times 1$ random vectors, $\boldsymbol{a}$ a conformable constant vector, and $\boldsymbol{A}$ and $\boldsymbol{B}$ are conformable constant matrices, then
$$E(\boldsymbol{a}+\boldsymbol{X})=\boldsymbol{a}+E(\boldsymbol{X}) \text { and } E(\boldsymbol{X}+\boldsymbol{Y})=E(\boldsymbol{X})+E(\boldsymbol{Y})$$
and
$$E(\boldsymbol{A} \boldsymbol{X})=\boldsymbol{A} E(\boldsymbol{X}) \text { and } E(\boldsymbol{A} \boldsymbol{X} \boldsymbol{B})=\boldsymbol{A} E(\boldsymbol{X}) \boldsymbol{B} .$$
Thus
$$\operatorname{Cov}(\boldsymbol{a}+\boldsymbol{A} \boldsymbol{X})=\operatorname{Cov}(\boldsymbol{A} \boldsymbol{X})=\boldsymbol{A} \operatorname{Cov}(\boldsymbol{X}) \boldsymbol{A}^T .$$
Some important properties of multivariate normal (MVN) distributions are given in the following three propositions. These propositions can be proved using results from Johnson and Wichern (Johnson and Wichern (1988), pp. $127-132)$

## 统计代写|线性回归代写线性回归代考|多元模型

10.1.

$$f(\boldsymbol{z} \mid \boldsymbol{\mu}, \boldsymbol{\Sigma})$$

## 统计代写|线性回归代写线性回归代考|多元正态分布

$$f(\boldsymbol{z})=\frac{1}{(2 \pi)^{p / 2}|\boldsymbol{\Sigma}|^{1 / 2}} e^{-(1 / 2)\left(\boldsymbol{z}{-} \boldsymbol{\mu}\right)^T \boldsymbol{\Sigma}^{-1}\left(\boldsymbol{z}{-} \boldsymbol{\mu}\right)}$$
where $|\Sigma|^{1 / 2}$ 行列式的平方根是多少 $\boldsymbol{\Sigma}$。注意，如果 $p=1$，则指数中的二次型为 $(z-\mu)\left(\sigma^2\right)^{-1}(z-\mu)$ 和 $X$ 有单变量 $N\left(\mu, \sigma^2\right)$ pdf。如果 $\Sigma$ 是正半定而非正定吗 $\boldsymbol{X}$ 具有简并分布。例如，单变量 $N\left(0,0^2\right)$ 分布是简并的(点质量在0处)

10.3.

$$E(\boldsymbol{X})=\left(E\left(X_1\right), \ldots, E\left(X_p\right)\right)^T$$

$$\operatorname{Cov}(\boldsymbol{X})=E(\boldsymbol{X}-E(\boldsymbol{X}))(\boldsymbol{X}-E(\boldsymbol{X}))^T=\left(\sigma_{i j}\right) .$$

$$E(\boldsymbol{a}+\boldsymbol{X})=\boldsymbol{a}+E(\boldsymbol{X}) \text { and } E(\boldsymbol{X}+\boldsymbol{Y})=E(\boldsymbol{X})+E(\boldsymbol{Y})$$

$$E(\boldsymbol{A} \boldsymbol{X})=\boldsymbol{A} E(\boldsymbol{X}) \text { and } E(\boldsymbol{A} \boldsymbol{X} \boldsymbol{B})=\boldsymbol{A} E(\boldsymbol{X}) \boldsymbol{B} .$$

$$\operatorname{Cov}(\boldsymbol{a}+\boldsymbol{A} \boldsymbol{X})=\operatorname{Cov}(\boldsymbol{A} \boldsymbol{X})=\boldsymbol{A} \operatorname{Cov}(\boldsymbol{X}) \boldsymbol{A}^T .$$以下三个命题给出了多元正态分布的一些重要性质。这些命题可以用约翰逊和威彻恩(约翰逊和威彻恩(1988))的结果来证明。 $127-132)$

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