[6] a random array of normal

[6] a random array of normal

对于一个矩阵
\[ X= \left( \begin{array} {cccc} x_{11} & x_{12} & \dots & x_{1p}\\ x_{21} & x_{22} & \dots & x_{2p}\\ \vdots & \vdots & & \vdots \\ x_{n1} & x_{n2} & \dots & x_{np}\\ \end{array} \right)= \left( \begin{array} {c} X'_{(1)}\\ X'_{(2)}\\ \vdots\\ X'_{(n)} \end{array} \right)=(\mathcal{X}_1,\mathcal{X}_2\dots,\mathcal{X}_p) \]

Set \ (X-_ {(I)} = (X_ {I1}, \ DOTS, X_ {IP}) '\) , ( \ (I =. 1, \ DOTS, n-\) ) is derived from \ (P \) Element normal population \ (N_p (\ mu, \ Sigma) \) is independent and identically distributed random samples, denoted random array \ (X-= (X_ {ij of}) _ {n-\ P} Times \) , using the straightening operation , \ ((\ mathbb {I} :: = p -dimensional unit vector) \) and a Kronecker product (the Kronecker) operation, found:
\ [Vec (X-') \ {NP} SIM of N_ (\ mathbb the I {} _n \ otimes \ mu, I_n \ otimes \ Sigma) \]

In fact,
\ [Vec (X-') = \ left (\ Array the begin {C} {} {X-_ (. 1)} \\ \ \\ X-vdots _ {(n-)} \} End {Array \ right) = ( x_ {11}, \ dots,
x_ {1p}, \ dots, x_ {n1}, \ dots, x_ {np}) '\] as a \ (NP \) long vector dimension, which is the joint density function:
\[ \begin{align} f(x_{(1)},\dots,x_{(n)}) =&\prod_{i=1}^n\frac1{(2\pi)^{p/2}|\Sigma|^{1/2}}exp\{-\frac12(x_{(i)}-\mu)'\Sigma^{-1}(x_{(i)}-\mu)\}\\ =&\frac1{(2\pi)^{np/2}|\Sigma|^{n/2}}exp\{-\frac12\sum_{i=1}^n(x_{(i)}-\mu)'\Sigma^{-1}(x_{(i)}-\mu)\}\\ =&\frac1{(2\pi)^{np/2}|\Sigma|^{n/2}}exp\left\{-\frac12\left( \begin{array}{c} x_{(1)}-\mu\\ \vdots\\ x_{(n)}-\mu \end{array} \right)' \left( \begin{array}{ccc} \Sigma&\cdots&O\\ \vdots&&\vdots\\ O&\cdots&\Sigma\\ \end{array} \right)^{-1} \left( \begin{array}{c} x_{(1)}-\mu\\ \vdots\\ x_{(n)}-\mu \end{array} \right) \right\}\\ =&\frac1{(2\pi)^{np/2}|\Sigma|^{n/2}}exp\left\{-\frac12\left( \begin{array}{c} X-\mathbb{I}_n\otimes\mu \end{array} \right)'(I_n \ otimes \ Sigma) ^ {- 1} (X- \ mathbb {I} _n \ otimes \ mu) \ right \} \\ \ end {align} \] by\ (X-\)
Thus, when the random arrayRow after straightening, if yes \ (Vec (X-') \ {NP} SIM of N_ (\ mathbb the I {} _n \ otimes \ MU, I_n \ otimes \ Sigma) \) , which is said to obey the matrix normal distribution , referred to as: \ (X-\ {n-SIM of N_ \} P Times (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
\[ Vec(X')\sim N_{np}(Vec(M'),I_n\otimes\Sigma)\quad\leftrightarrows\quad X\sim N_{n\times p}(M，I_n\otimes\Sigma) \]

Properties of the linear combination

• 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} (WITH 'D + (AA) \ otimes (B \ Sigma B')) \]