Wishart distribution

: Multivariate extension of $\chi^2$ distribution

• pdf

  $f(w)=\frac{det(w)^{(r-p-1)/2}}{2^{rp/2} \Gamma_p(r/2) det(\Sigma)^{r/2}} exp(-\frac{1}{2} tr(\Sigma^{-1}w))$
  for $w \in \mathbb{R}^{p \times p}: \text{positive definite}, r>p-1$ and $\Sigma \in \mathbb{R}^{p \times p}: \text{positive definite}$

• 동치명제

  $W \sim W_p(r,\Sigma),W \in \mathbb{R}^{p \times p}$
  <=> $W \overset{d}{\equiv} \sum_{j=1}^{r} Z_j Z_j^T \text{ where } Z_j \overset{iid}{\sim} N_p(0,\Sigma)(j=1,\cdots,r)$
  <=> For any $d \in \mathbb{R}^p, d^T W d \sim d^T \Sigma d \chi^2_r$

• Properties

  ① Suppose $X_1,\cdots,X_n \overset{iid}{\sim} N_p(0,\Sigma), X=(X_1,\cdots,X_n)^T \in \mathbb{R}^{n \times p} \text{ and } A \in \mathbb{R}^{n \times n}: \text{symmtric}$
  If $A^2=A$ and $r=tr(A)$, we have $X^T A X \sim W_p(r,\Sigma)$
  
  ② If $W_1 \sim W_p(r_1,\Sigma),W_2 \sim W_p(r_2,\Sigma) \text{ and } W_1 \perp W_2$, then $W_1 + W_2 \sim W_p(r_1+r_2, \Sigma)$
  
  ③ If $W \sim W_p(r, \Sigma)$, then $CWC^T \sim W_q(r, C \Sigma C^T)$ for any $C \in \mathbb{R}^{q \times p}$ with $rank(C)=q \leq p$
  
  ④ If $W=(w_{ij}) \sim W_p(r,\Sigma) \text{ where } \Sigma=(\sigma_{ij})$, then $E(w)=r \Sigma$ and $Var(w_{ij})=r(\sigma^2_{ij}+\sigma_{ii}\sigma_{jj})$
  pf)
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