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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|应用线性模型代写Applied Linear Models代考|POSITIVE DEFINITENESS

A property of some quadratic forms used repeatedly in what follows is that of positive definiteness. A quadratic form $\mathbf{x}^{\prime} \mathbf{A x}$ is said to be positive definite if it is positive for all values of $\mathbf{x}$ except $\mathbf{x}=\mathbf{0}$; i.e., if
$$\mathbf{x}^{\prime} \mathbf{A x}>0 \quad \text { for all } \mathbf{x} \text {, except } \mathbf{x}=\mathbf{0},$$
then $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}$ is positive definite. And the corresponding (symmetric) matrix is also described as positive definite.
Example.
\begin{aligned} \mathbf{x}^{\prime} \mathbf{A x} &=\left[\begin{array}{lll} x_1 & x_2 & x_3 \end{array}\right]\left[\begin{array}{rrr} 3 & 5 & 1 \ 5 & 13 & 0 \ 1 & 0 & 1 \end{array}\right]\left[\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right] \ &=3 x_1^2+13 x_2^2+x_3^2+10 x_1 x_2+2 x_1 x_3 \end{aligned}

can be rearranged as
$$\mathbf{x}^{\prime} \mathbf{A x}=\left(x_1+2 x_2\right)^2+\left(x_1+3 x_2\right)^2+\left(x_1+x_3\right)^2$$
which is positive for any (real) values of the $x$ ‘s except $x_1=0=x_2=x_3$, i.e., except for $\mathbf{x}=\mathbf{0}$ (in which case $\mathbf{x}^{\prime} \mathbf{A x}$ is always zero). Hence $\mathbf{x}^{\prime} \mathbf{A x}$ is positive definite (abbreviated p.d.).

A slight relaxation of the above definition concerns $\mathbf{x}^{\prime} \mathbf{A x}$ when its value is either positive or zero for all $\mathbf{x} \neq \mathbf{0}$. We define an $\mathbf{x}^{\prime} \mathbf{A x}$ of this nature as being positive semi-definite (abbreviated p.s.d.) when
$\mathbf{x}^{\prime} \mathbf{A x} \geq 0 \quad$ for all $\mathbf{x} \neq \mathbf{0}$, with $\mathbf{x}^{\prime} \mathbf{A x}=0$ for at least one $\mathbf{x} \neq \mathbf{0}$.
Under these conditions $\mathbf{x}^{\prime} \mathbf{A x}$ is a p.s.d. quadratic form and the corresponding symmetric matrix $\mathbf{A}$ is a p.s.d. matrix. This definition is widely accepted [e.g., Graybill (1961, p. 3) and Rao (1965, p. 31)], although not universally so. For example, a definition used by Scheffé (1959, p. 398) is that $\mathbf{A}$ is a p.s.d. matrix when $\mathbf{x}^{\prime} \mathbf{A x} \geq 0$ for all $\mathbf{x} \neq 0$ without demanding that $\mathbf{x}^{\prime} \mathbf{A x}=0$ for at least one non-null $\mathbf{x}$. Hence this definition includes matrices that we have defined as either p.d. or p.s.d. We will call such matrices non-negative definite (n.n.d.) in keeping, for example, with Rao (1965, p. 31). Thus n.n.d. matrices are either p.d. or p.s.d.

## 统计代写|应用线性模型代写Applied Linear Models代考|DISTRIBUTIONS

For the sake of reference and establishing notation, certain salient features of commonly used statistical distributions are now summarized. No attempt is made at completeness or full rigor. Any number of texts [e.g., Graybill (1961), Wilks (1962), Mood and Graybill (1963) and Hogg and Craig (1965)] give the pertinent details with which, it is assumed, the reader will be familiar. What follows will serve only to remind him of these things.
a. Multivariate density functions
In considering $n$ random variables $X_1, X_2, \ldots, X_n$, for which $x_1, x_2, \ldots$, $x_n$ represents a set of realized values we write the cumulative density function as
$$\operatorname{Pr}\left(X_1 \leq x_1, X_2 \leq x_2, \ldots, X_n \leq x_n\right)=F\left(x_1, x_2, \ldots, x_n\right) .$$
Then the density function is
$$f\left(x_1, x_2, \ldots, x_n\right)=\frac{\partial^n}{\partial x_1 \partial x_2 \ldots \partial x_n} F\left(x_1, x_2, \ldots, x_n\right) .$$
Conditions which $f\left(x_1, x_2, \ldots, x_n\right)$ must satisfy are $\quad f\left(x_1, x_2, \ldots, x_n\right) \geq 0$ for $-\infty<x_i<\infty$ for all $i$
$$\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_1, x_2, \ldots, x_n\right) d x_1 d x_2 \ldots d x_n=1 \text {. }$$
The marginal density function for what might be called the “last $n-k x$ ‘s” is $f\left(x_1, x_2, \ldots, x_n\right)$ after integrating out the first $k x$ ‘s, i.e., the marginal of $x_{k+1}, \ldots, x_n$ is
$$g\left(x_{k+1}, \ldots, x_n\right)=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_1, \ldots, x_k, x_{k+1}, \ldots, x_n\right) d x_1 \ldots d x_k$$

# 应用线性模型代考

## 统计代写|应用线性模型代写Applied Linear Models代考|POSITIVE DEFINITENESS

$$\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}>0 \quad \text { for all } \mathbf{x}, \text { except } \mathbf{x}=\mathbf{0},$$

$$\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}=\left(x_1+2 x_2\right)^2+\left(x_1+3 x_2\right)^2+\left(x_1+x_3\right)^2$$

## 统计代写|应用线性模型代写Applied Linear Models代考|DISTRIBUTIONS

$$\operatorname{Pr}\left(X_1 \leq x_1, X_2 \leq x_2, \ldots, X_n \leq x_n\right)=F\left(x_1, x_2, \ldots, x_n\right) .$$

$$f\left(x_1, x_2, \ldots, x_n\right)=\frac{\partial^n}{\partial x_1 \partial x_2 \ldots \partial x_n} F\left(x_1, x_2, \ldots, x_n\right) .$$

$$\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} f\left(x_1, x_2, \ldots, x_n\right) d x_1 d x_2 \ldots d x_n=1 .$$

$$g\left(x_{k+1}, \ldots, x_n\right)=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_1, \ldots, x_k, x_{k+1}, \ldots, x_n\right) d x_1 \ldots d x_k$$

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assignmentutor™您的专属作业导师
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