توزيع ديريشليه للمتغيرات المصفوفية

يفترض${\displaystyle S_{i}\sim W_{p}\left(n_{i},\Sigma \right),i=1,\ldots ,r+1}$يتم توزيعها بشكل مستقل بواسطة شركة ويشارت${\displaystyle p\times p}$المصفوفات الموجبة المحددة . ثم، تعريف${\displaystyle U_{i}=S^{-1/2}S_{i}\left(S^{-1/2}\right)^{T}}$(أين${\displaystyle S=\sum _{i=1}^{r+1}S_{i}}$هو مجموع المصفوفات و${\displaystyle S^{1/2}\left(S^{-1/2}\right)^{T}}$أي تحليل معقول لـ${\displaystyle S}$لدينا

${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(n_{1}/2,...,n_{r+1}/2\right).}$

لو${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r+1}\right)}$وإذا${\displaystyle s\leq r}$، ثم:

${\displaystyle \left(U_{1},\ldots ,U_{s}\right)\sim D_{p}\left(a_{1},\ldots ,a_{s},\sum _{i=s+1}^{r+1}a_{i}\right)}$

كذلك، وباستخدام نفس الرموز المذكورة أعلاه، فإن كثافة${\displaystyle \left(U_{s+1},\ldots ,U_{r}\right)\left|\left(U_{1},\ldots ,U_{s}\right)\right.}$يُعطى بواسطة

${\displaystyle {\frac {\prod _{i=s+1}^{r+1}\det \left(U_{i}\right)^{a_{i}-(p+1)/2}}{\beta _{p}\left(a_{s+1},\ldots ,a_{r+1}\right)\det \left(I_{p}-\sum _{i=1}^{s}U_{i}\right)^{\sum _{i=s+1}^{r+1}a_{i}-(p+1)/2}}}}$

حيث نكتب${\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{i}}$.

يفترض${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r+1}\right)}$ولنفترض أن${\displaystyle S_{1},\ldots ,S_{t}}$هو تقسيم لـ${\displaystyle \left[r+1\right]=\left\{1,\ldots r+1\right\}}$(إنه،${\displaystyle \cup _{i=1}^{t}S_{i}=\left[r+1\right]}$و${\displaystyle S_{i}\cap S_{j}=\emptyset }$لو${\displaystyle i\neq j}$ثم الكتابة${\displaystyle U_{(j)}=\sum _{i\in S_{j}}U_{i}}$و${\displaystyle a_{(j)}=\sum _{i\in S_{j}}a_{i}}$(مع${\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{r}}$لدينا:

${\displaystyle \left(U_{(1)},\ldots U_{(t)}\right)\sim D_{p}\left(a_{(1)},\ldots ,a_{(t)}\right).}$

يفترض${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r+1}\right)}$. يُعرِّف

${\displaystyle U_{i}=\left({\begin{array}{rr}U_{11(i)}&U_{12(i)}\\U_{21(i)}&U_{22(i)}\end{array}}\right)\qquad i=1,\ldots ,r}$

أين${\displaystyle U_{11(i)}}$يكون${\displaystyle p_{1}\times p_{1}}$و ${\displaystyle U_{22(i)}}$يكون${\displaystyle p_{2}\times p_{2}}$كتابة مكمل شور${\displaystyle U_{22\cdot 1(i)}=U_{21(i)}U_{11(i)}^{-1}U_{12(i)}}$لدينا

${\displaystyle \left(U_{11(1)},\ldots ,U_{11(r)}\right)\sim D_{p_{1}}\left(a_{1},\ldots ,a_{r+1}\right)}$

و

${\displaystyle \left(U_{22.1(1)},\ldots ,U_{22.1(r)}\right)\sim D_{p_{2}}\left(a_{1}-p_{1}/2,\ldots ,a_{r}-p_{1}/2,a_{r+1}-p_{1}/2+p_{1}r/2\right).}$