多元正态分布的一些性质

Posted 2021-03-08 rrrrraulista

• 若将(Xsim N_p(mu,Sigma))进行分割：
  [ X= left[ egin{array}{c} X^{(1)}_rX^{(2)}_{p-r} end{array} ight]， mu= left[ egin{array}{c} mu^{(1)}_r\mu^{(2)}_{p-r} end{array} ight]， Sigma= left[ egin{array}{c|c} Sigma_{11} &Sigma_{12}\ hline Sigma_{21} &Sigma_{22} end{array} ight]>0，(Sigma_{11}为r imes r方阵) ]

一、独立性

设 (p) 维随机向量 (Xsim N_p(mu,Sigma)),
[ X= left[ egin{array}{c} X^{(1)}X^{(2)} end{array} ight]sim left( left[ egin{array}{c} mu^{(1)}\mu^{(2)} end{array} ight]， left[ egin{array}{cc} Sigma_{11} &Sigma_{12}\ Sigma_{21} &Sigma_{22} end{array} ight] ight) ]
则
[ X^{(1)}与 X^{(2)}相互独立 leftrightarrows Sigma_{12}=O ]

(证明)

设(Sigma_{12}=O)，则(X)的联合密度函数为：
[ egin{align} f(x^{(1)},x^{(2)})=& frac1{(2pi)^{p/2}|Sigma|^{1/2}}expleft(-frac12(x-mu)' left[ egin{array}{cc} Sigma_{11}&OO&Sigma_{22} end{array} ight]^{-1} (x-mu) ight)=& frac1{(2pi)^{r/2}|Sigma_{11}|^{1/2}}expleft(-frac12(x^{(1)}-mu^{(1)})' Sigma_{11}^{-1} (x^{(1)}-mu^{(1)}) ight)&cdot frac1{(2pi)^{(p-r)/2}|Sigma_{22}|^{1/2}}expleft(-frac12(x^{(2)}-mu^{(2)})' Sigma_{22}^{-1} (x^{(2)}-mu^{(2)}) ight)=&f_1(x^{(1)})cdot f_2(x^{(2)}) end{align} ]

因此(X^{(1)},X^{(2)})相互独立。

• 设(r_igeq1,(i=1,dots,k)),且(r_1+r_2+dots+r_k=p),则有

[ X= left[ egin{array}{c} X^{(1)}\vdotsX^{(k)} end{array} ight]sim N_p left( left[ egin{array}{c} mu^{(1)}\vdots\mu^{(k)} end{array} ight]， left[ egin{array}{ccc} Sigma_{11} &cdots &Sigma_{1k}\ vdots&&vdots\Sigma_{k1} &cdots &Sigma_{kk} end{array} ight]_{p imes p} ight) ]

则(X^{(1)},X^{(2)},dots,X^{(k)})相互独立 (leftrightarrows) (Sigma_{ij}=O,(i eq j)).

• 设(X=(X_1,dots,X_p)'sim N_p(mu,Sigma)),若(Sigma)为对角矩阵，则(X_1,dots,X_p)相互独立。

二、条件分布

当(X_2)给定时，(X_1)的条件密度为：
[ f_1(x_1|x_2)=frac{f(x_1,x_2)}{f_2(x_2)} ]

[ egin{align} f(x_1,x_2)= &=frac{1}{2pisigma_1sigma_2sqrt{1- ho^2}}expleft{-frac{1}{2(1- ho^2)}[(frac{x_1-mu_1}{sigma_1})^2-2 ho(frac{x_1-mu_1}{sigma_1})(frac{x_2-mu_2}{sigma_2})+(frac{x_2-mu_2}{sigma_2})^2] ight}&=frac{1}{2pisigma_1sigma_2sqrt{1- ho^2}}expleft{-frac{1}{2(1- ho^2)}[(frac{x_1-mu_1}{sigma_1})^2-2 ho(frac{x_1-mu_1}{sigma_1})(frac{x_2-mu_2}{sigma_2})+(1- ho^2)(frac{x_2-mu_2}{sigma_2})^2+ ho^2(frac{x_2-mu_2}{sigma_2})^2] ight}&=frac{1}{2pisigma_1sigma_2sqrt{1- ho^2}}expleft{-frac{1}{2}(frac{x_2-mu_2}{sigma_2})^2 ight}cdot expleft{-frac{1}{2(1- ho^2)}[(frac{x_1-mu_1}{sigma_1})^2-2 ho(frac{x_1-mu_1}{sigma_1})(frac{x_2-mu_2}{sigma_2})+ ho^2(frac{x_2-mu_2}{sigma_2})^2] ight}&=frac{1}{2pisigma_1sigma_2sqrt{1- ho^2}}expleft{-frac{1}{2}(frac{x_2-mu_2}{sigma_2})^2 ight}cdot expleft{-frac{1}{2(1- ho^2)}[(frac{x_1-mu_1}{sigma_1})- ho(frac{x_2-mu_2}{sigma_2})]^2 ight}&=frac{1}{sqrt{2pi}sigma_2}expleft{-frac{1}{2}(frac{x_2-mu_2}{sigma_2})^2 ight}cdotfrac{1}{sqrt{2pi}sigma_1sqrt{1- ho^2}}cdot expleft{-frac{1}{2(1- ho^2)sigma_1^2}[x_1-mu_1- hofrac{sigma_1}{sigma_2}(x_2-mu_2)]^2 ight}&=f_2(x_2)cdot f(x_1|x_2) end{align} ]

其中

[ f(x_1|x_2)=frac{1}{sqrt{2pi}sigma_1sqrt{1- ho^2}}cdot expleft{ -frac{1}{2(1- ho^2)sigma_1^2}[x_1-left(mu_1 + hofrac{sigma_1}{sigma_2}(x_2-mu_2) ight)]^2 ight}\]

由定义：

[ (X_1|X_2)sim N_1left(mu_1+ hofrac{sigma_1}{sigma_2}(x_2-mu_2)，sigma^2(1- ho^2) ight) ]

将其推广到多维：

设
[ X= left[ egin{array}{c} X^{(1)}_rX^{(2)}_{p-r} end{array} ight]sim N_p(mu,Sigma)，(Sigma>0) ]
则当(X^{(2)})给定时，(X^{(1)})的条件分布为：
[ (X^{(1)}|X^{(2)})sim N_r(mu_{1cdot2}，Sigma_{11cdot2}) ]
其中
[ mu_{1cdot2}=mu^{(1)}+Sigma_{12}Sigma_{22}^{-1}(x^{(2)}-mu^{(2)})\Sigma_{11cdot2}=Sigma_{11}-Sigma_{12}Sigma_{22}^{-1}Sigma_{21} ]