【6】随机阵的正态分布
【6】随机阵的正态分布
对于一个矩阵
\[X=
\left(
\begin{array}
{cccc}
x_{11} & x_{12} & \dots & x_{1p}\\
x_{21} & x_{22} & \dots & x_{2p}\\
\vdots & \vdots & & \vdots \\
x_{n1} & x_{n2} & \dots & x_{np}\\
\end{array}
\right)=
\left(
\begin{array}
{c}
X'_{(1)}\\
X'_{(2)}\\
\vdots\\
X'_{(n)}
\end{array}
\right)=(\mathcal{X}_1,\mathcal{X}_2\dots,\mathcal{X}_p)
\]
设\(X_{(i)}=(x_{i1},\dots,x_{ip})'\),（\(i=1,\dots,n\)）为来自\(p\)元正态总体 \(N_p(\mu,\Sigma)\) 的独立同分布随机样本，记随机阵\(X=(x_{ij})_{n\times p}\)，利用拉直运算,\((\mathbb{I}::=p维单位向量)\)及克罗内克积( Kronecker )运算，可知：
\[Vec(X')\sim N_{np}(\mathbb{I}_n\otimes\mu,I_n\otimes\Sigma)
\]
事实上，
\[Vec(X')= \left( \begin{array}{c} X_{(1)}\\ \vdots\\ X_{(n)} \end{array} \right)=(x_{11},\dots,x_{1p},\dots,x_{n1},\dots,x_{np})' \]为一个\(np\)维的长向量，其联合密度函数为：
\[\begin{align} f(x_{(1)},\dots,x_{(n)}) =&\prod_{i=1}^n\frac1{(2\pi)^{p/2}|\Sigma|^{1/2}}exp\{-\frac12(x_{(i)}-\mu)'\Sigma^{-1}(x_{(i)}-\mu)\}\\ =&\frac1{(2\pi)^{np/2}|\Sigma|^{n/2}}exp\{-\frac12\sum_{i=1}^n(x_{(i)}-\mu)'\Sigma^{-1}(x_{(i)}-\mu)\}\\ =&\frac1{(2\pi)^{np/2}|\Sigma|^{n/2}}exp\left\{-\frac12\left( \begin{array}{c} x_{(1)}-\mu\\ \vdots\\ x_{(n)}-\mu \end{array} \right)' \left( \begin{array}{ccc} \Sigma&\cdots&O\\ \vdots&&\vdots\\ O&\cdots&\Sigma\\ \end{array} \right)^{-1} \left( \begin{array}{c} x_{(1)}-\mu\\ \vdots\\ x_{(n)}-\mu \end{array} \right) \right\}\\ =&\frac1{(2\pi)^{np/2}|\Sigma|^{n/2}}exp\left\{-\frac12\left( \begin{array}{c} X-\mathbb{I}_n\otimes\mu \end{array} \right)' (I_n\otimes\Sigma)^{-1} (X-\mathbb{I}_n\otimes\mu) \right\}\\ \end{align} \]于是，当随机阵\(X\)按行进行拉直以后，若满足\(Vec(X')\sim N_{np}(\mathbb{I}_n\otimes\mu,I_n\otimes\Sigma)\)，则称其服从矩阵正态分布，记作：\(X\sim N_{n\times p}(M，I_n\otimes\Sigma)\)
其中
\[M=\left( \begin{array} {ccc} \mu_1 & \dots & \mu_p\\ \vdots & & \vdots \\ \mu_1 & \dots & \mu_p\\ \end{array} \right) =\mathbb{I}_n\mu'::= \left( \begin{array} {c} 1\\\vdots\\1 \end{array} \right)_{p\times1} (\mu_1,\dots,\mu_p) \]则有
\[Vec(M')=\mathbb{I}_n\mu=(\mu_1,\dots,\mu_p,\dots,\mu_1,\dots,\mu_p)' \]于是
\[Vec(X')\sim N_{np}(Vec(M'),I_n\otimes\Sigma)\quad\leftrightarrows\quad X\sim N_{n\times p}(M，I_n\otimes\Sigma) \]
线性组合的性质
- 设\(X\sim N_{n\times p}(M，I_n\otimes\Sigma)\),令\(Z=A_{k\times n}XB_{q\times p}'+D_{k\times q}\)，则：
\[Z\sim N_{k\times q}(AMB'+D，(AA')\otimes(B\Sigma B'))
\]
