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多元统计分析：Wishart分布的小记

Wishart分布的正定性

设\mathbf{M}\sim W _ p(n,\Sigma,\Delta),\; n\geqslant p,\; \Sigma>0，则有
P\, (\mathbf{M}>0)=1.

证明：

应有\mathbf{M}=\mathbf{X^\prime X},\; \mathbf{X}\sim N _ {n,p}(M,\Sigma,I),\; M^\prime M=\Delta.

记\mathbf{X}=(\mathbf{x} _ {(1)},\dots,\mathbf{x} _ {(p)})，对k<p，由正态分布条件分布性质，当\mathbf{x} _ {(1)},\dots,\mathbf{x} _ {(k)}给定时，\mathbf{x} _ {(k+1)}的条件分布是N _ p(\mathrm{v},\sigma _ {k+1}^2 I)，可以推出\mathbf{x} _ {(k+1)}的条件分布非退化，故有

P\left(\mathbf{x} _ {(k+1)} \in \mu\left(\mathbf{x} _ {(1)}, \cdots, \mathbf{x} _ {(k)}\right) \mid \mathbf{x} _ {(1)}, \cdots, \mathbf{x} _ {(k)}\right)=0.

这是由于\mu\left(\mathbf{x} _ {(1)}, \cdots, \mathbf{x} _ {(k)}\right)至多是k维子空间，而\mathbf{x} _ {(k+1)}是n维的，且条件分布非退化，从而有

\begin{aligned}
&\phantom{=}P\left(\mathbf x _ {(k+1)} \in \mu\left(\mathbf x _ {(1)}, \cdots, \mathbf x _ {(k)}\right)\right)\\
&=E\left\{P\left(\mathbf x _ {(k+1)} \in \mu\left(\mathbf x _ {(1)}, \cdots, \mathbf x _ {(k)}\right) \mid \mathbf x _ {(1)}, \cdots, \mathbf x _ {(k)}\right)\right\}
\&=0.
\end{aligned}

于是
\begin{aligned}
&\phantom{=}P(\mathrm{rank}\mathbf{X}<p)\\
&=P(\mathbf x _ {(1)}, \cdots, \mathbf x _ {(p)}\text{线性相关})\\
&\leqslant\sum _ {k=0}^{p-1} P(\mathbf x _ {(k+1)} \in \mu\left(\mathbf x _ {(1)}, \cdots, \mathbf x _ {(k)}\right)\\
&=0.
\end{aligned}

因此有P\, (\mathbf{X^\prime X}>0)=1，即证.

Wishart分布一个性质的证明

定理：设\mathbf{M}\sim W _ p(m,\Sigma),\; m\geqslant p,\; \Sigma>0，并记
\mathbf{M}=
\begin{pmatrix}
\mathbf{M} _ {11}&\mathbf{M} _ {12}\\
\mathbf{M} _ {21}&\mathbf{M} _ {22}
\end{pmatrix},\quad
\Sigma=
\begin{pmatrix}
\Sigma _ {11}&\Sigma _ {12} \\ \Sigma _ {21}&\Sigma _ {22}
\end{pmatrix},

其中\mathbf{M} _ {11}为a\times a阶的，\mathbf{M} _ {22}为b\times b阶的，a+b=p，并记
\mathbf{M} _ {2 \cdot 1}=\mathbf{M} _ {22}-\mathbf{M} _ {21} \mathbf{M} _ {11}^{-1} \mathbf{M} _ {12},\quad
\Sigma _ {2 \cdot 1}=\Sigma _ {22}-\Sigma _ {21} \Sigma _ {11}^{-1} \Sigma _ {12}.

那么，在给定\mathbf{M} _ {11}下，\mathbf{M} _ {12} \sim N _ {a,b}\left(\mathbf{M} _ {11}\Sigma _ {11}^{-1}\Sigma _ {12} , \Sigma _ {2 \cdot 1}, \mathbf{M} _ {11}\right).

（出自见张润楚编著《多元统计分析》第40页，沿用原书记号，原书给出的是\mathbf{M} _ {21}的分布，没有给出证明，怀疑是错的）

证明：

设\mathbf{M=X^\prime X,\; X}\sim N _ {n,p}(\mathbf{0},\Sigma,I)，记\mathbf{X}=(\mathbf{X} _ 1,\mathbf{X} _ 2)，这里\mathbf{X} _ 1有a列，\mathbf{X} _ 2有b列，那么

\mathbf{X}^{\mathrm{V}}=
\begin{pmatrix}
\mathbf{X} _ 1^{\mathrm{V}}\\
\mathbf{X} _ 2^{\mathrm{V}}
\end{pmatrix},\quad
\mathrm{Cov}(\mathbf{X}^{\mathrm{V}})=\Sigma\otimes I=
\begin{pmatrix}
\Sigma _ {11}\otimes I&\Sigma _ {12}\otimes I \\
\Sigma _ {21}\otimes I&\Sigma _ {22}\otimes I
\end{pmatrix}.

于是
\mathbf{X} _ 1^{\mathrm{V}}\sim N _ {na}(\, \mathbf{0},\; \Sigma _ {11}\otimes I\, ),\; \mathbf{X} _ 2^{\mathrm{V}}\sim N _ {nb}(\, \mathbf{0},\; \Sigma _ {22}\otimes I\, ),\; \mathrm{Cov}(\mathbf{X} _ 1^{\mathrm{V}},\mathbf{X} _ 2^{\mathrm{V}})=\Sigma _ {12}\otimes I.

在\mathbf{X} _ 1给定时，有
\begin{gathered}
E(\mathbf{X} _ 2^{\mathrm{V}}|\mathbf{X} _ 1^{\mathrm{V}})=(\Sigma _ {21}\otimes I)(\Sigma _ {11}^{-1}\otimes I)\mathbf{X} _ 1^{\mathrm{V}}=(\Sigma _ {21}\Sigma _ {11}^{-1}\otimes I)\mathbf{X} _ 1^{\mathrm{V}}=(\mathbf{X} _ 1\Sigma _ {11}^{-1}\Sigma _ {12})^{\mathrm{V}},\\
\mathrm{Cov}(\mathbf{X} _ 2^{\mathrm{V}}|\mathbf{X} _ 1^{\mathrm{V}})=\Sigma _ {22}\otimes I-(\Sigma _ {21}\otimes I)(\Sigma _ {11}\otimes I)^{-1}(\Sigma _ {12}\otimes I)=\Sigma _ {2.1}\otimes I.
\end{gathered}
故
\mathbf{X} _ 2^{\mathrm{V}}|\mathbf{X} _ 1^{\mathrm{V}}\sim N _ {nb}
(\, (\mathbf{X} _ 1\Sigma _ {11}^{-1}\Sigma _ {12})^{\mathrm{V}},\, \Sigma _ {2.1}\otimes I\, )
由此不难得到在\mathbf{X} _ 1给定时\mathbf{X} _ 2的条件分布为
\mathbf{X} _ {2}|\mathbf{X} _ {1} \sim N _ {n,b}\left(\mathbf{X} _ {1} {\Sigma} _ {11}^{-1} {\Sigma} _ {12}, {\Sigma} _ {2.1}, {I}\right).

注意到\mathbf{M} _ {11}=\mathbf{X}^\prime _ 1\mathbf{X} _ 1,\; \mathbf{M} _ {22}=\mathbf{X}^\prime _ 2\mathbf{X} _ 2,\; \mathbf{M} _ {12}=\mathbf{X}^\prime _ 1\mathbf{X} _ 2，当\mathbf{M} _ {11}给定时\mathbf{X} _ 1也给定，故\mathbf{M} _ {12}^{\mathrm{V}}=(\mathbf{X}^\prime _ 1\mathbf{X} _ 2)^{\mathrm{V}}=\left(I\otimes \mathbf{X}^\prime _ 1\right)\mathbf{X} _ 2^{\mathrm{V}}的条件分布为
\begin{aligned}
\mathbf{M} _ {12}^{\mathrm{V}}|\mathbf{M} _ {11}&\sim N _ {ab}\left(\left(I\otimes \mathbf{X}^\prime _ 1\right)(\mathbf{X} _ 1\Sigma _ {11}^{-1}\Sigma _ {12})^{\mathrm{V}},\, \left(I\otimes \mathbf{X}^\prime _ 1\right)(\Sigma _ {2.1}\otimes I)\left(I\otimes \mathbf{X} _ 1\right)\right)\\
&=N _ {ab}\left((\mathbf{M} _ {11}\Sigma _ {11}^{-1}\Sigma _ {12})^{\mathrm{V}},\, (\Sigma _ {2.1}\otimes \mathbf{M} _ {11})\right)
\end{aligned}
所以，
\mathbf{M} _ {12}|\mathbf{M} _ {11} \sim N _ {a,b}\left(\mathbf{M} _ {11}\Sigma _ {11}^{-1}\Sigma _ {12} , \Sigma _ {2 \cdot 1}, \mathbf{M} _ {11}\right).