Yilda statistika, matritsa o'zgaruvchan Dirichlet taqsimoti ning umumlashtirilishi matritsa o'zgaruvchan beta-taqsimot.
Aytaylik
bor
ijobiy aniq matritsalar bilan
, qayerda
bo'ladi
identifikatsiya matritsasi. Keyin biz aytamiz
matritsa o'zgaruvchan Dirichlet taqsimotiga ega,
, agar ularning qo'shma qismi ehtimollik zichligi funktsiyasi bu
![{ displaystyle left { beta _ {p} left (a_ {1}, ldots, a_ {r}, a_ {r + 1} right) right } ^ {- 1} prod _ {i = 1} ^ {r} det chap (U_ {i} o'ng) ^ {a_ {i} - (p + 1) / 2} det chap (I_ {p} - sum _ { i = 1} ^ {r} U_ {i} o'ng) ^ {a_ {r + 1} - (p + 1) / 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e46e75b0604681a8e86ba6c537f8d86460810fe)
qayerda
va
bo'ladi ko'p o'zgaruvchan beta-funktsiya.
Agar biz yozsak
keyin PDF oddiyroq shaklga ega bo'ladi
![{ displaystyle left { beta _ {p} left (a_ {1}, ldots, a_ {r + 1} right) right } ^ {- 1} prod _ {i = 1} ^ {r + 1} det chap (U_ {i} o'ng) ^ {a_ {i} - (p + 1) / 2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c35063d5bb6dda2e1aeb097ae211feb0442fad8)
buni tushunish bo'yicha
.
Teoremalar
chi kvadrat-Dirichlet natijasini umumlashtirish
Aytaylik
mustaqil ravishda tarqatiladi Tilak
ijobiy aniq matritsalar. Keyin, belgilash
(qayerda
matritsalarning yig'indisi va
ning har qanday oqilona faktorizatsiyasi
), bizda ... bor
![{ displaystyle chap (U_ {1}, ldots, U_ {r} o'ng) sim D_ {p} chap (n_ {1} / 2, ..., n_ {r + 1} / 2 ) o'ng).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ce2c19c55843c482678f24a535df2fdda88c53)
Marginal taqsimot
Agar
va agar bo'lsa
, keyin:
![{ displaystyle chap (U_ {1}, ldots, U_ {s} o'ng) sim D_ {p} chap (a_ {1}, ldots, a_ {s}, sum _ {i = s +1} ^ {r + 1} a_ {i} o'ng)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54154d885753f47b35ed8f9b372a5c0322ac7f9)
Shartli taqsimot
Bundan tashqari, yuqoridagi kabi yozuv bilan, ning zichligi
tomonidan berilgan
![{ displaystyle { frac { prod _ {i = s + 1} ^ {r + 1} det left (U_ {i} right) ^ {a_ {i} - (p + 1) / 2} } { beta _ {p} chap (a_ {s + 1}, ldots, a_ {r + 1} o'ng) det chap (I_ {p} - sum _ {i = 1} ^ { s} U_ {i} o'ng) ^ { sum _ {i = s + 1} ^ {r + 1} a_ {i} - (p + 1) / 2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/344da901ae33d0f9ebd479b4c1f554196d708c40)
qaerga yozamiz
.
taqsimlangan tarqatish
Aytaylik
va buni taxmin qiling
ning bo'limi
(anavi,
va
agar
). Keyin, yozish
va
(bilan
), bizda ... bor:
![{ displaystyle chap (U _ {(1)}, ldots U _ {(t)} right) sim D_ {p} left (a _ {(1)}, ldots, a _ {(t)} o'ng).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5281a346a3fd7718207dd8b4a3390ef98f9ab6)
bo'limlar
Aytaylik
. Aniqlang
![{ 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/210f8e9bdbc95eb591ac1e31437461406fe3a9df)
qayerda
bu
va
bu
. Yozish Schur to'ldiruvchisi
bizda ... bor
![{ displaystyle chap (U_ {11 (1)}, ldots, U_ {11 (r)} o'ng) sim D_ {p_ {1}} chap (a_ {1}, ldots, a_ {r +1} o'ng)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e945294eab29daad2451cafa40aba9edebde7ac7)
va
![{ displaystyle chap (U_ {22.1 (1)}, ldots, U_ {22.1 (r)} o'ng) sim D_ {p_ {2}} chap (a_ {1} -p_ {1} / 2) , ldots, a_ {r} -p_ {1} / 2, a_ {r + 1} -p_ {1} / 2 + p_ {1} r / 2 o'ng).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79befd7cf0722c9a6ff85193ab2e4e3f643a9923)
Shuningdek qarang
Adabiyotlar
A. K. Gupta va D. K. Nagar 1999. "Matritsaning turlicha taqsimlanishi". Chapman va Xoll.