Yilda statistika , xususan gipotezani sinash , Hotelling T - kvadrat taqsimot (T 2 Garold Hotelling ,[1] ko'p o'zgaruvchan ehtimollik taqsimoti bilan chambarchas bog'liq F - tarqatishstatistika namunalari bu asosda joylashgan statistik ma'lumotlarning tabiiy umumlashtirilishi Talaba t - tarqatish .
The Hotelling t - kvadrat statistikasi (t 2 Talaba t -statistik ichida ishlatiladigan ko'p o'zgaruvchan gipotezani sinash .[2]
Tarqatish
Motivatsiya Tarqatish paydo bo'ladi ko'p o'zgaruvchan statistika qabul qilishda testlar bir xil o'zgaruvchan muammolar uchun testlardan foydalanadigan turli xil populyatsiyalarning (ko'p o'zgaruvchan) vositalari o'rtasidagi farqlarning t -testGarold Hotelling , uni Talabaning umumlashtirilishi sifatida ishlab chiqqan t - tarqatish.[1]
Ta'rif Agar vektor bo'lsa d { displaystyle d} Gauss ko'p o'zgaruvchan taqsimlangan nolinchi o'rtacha va birlik bilan kovaryans matritsasi N ( 0 p , Men p , p ) { displaystyle N ( mathbf {0} _ {p}, mathbf {I} _ {p, p})} M { displaystyle M} p × p { displaystyle p times p} o'lchov matritsasi va m erkinlik darajasi bilan Istaklarni tarqatish V ( Men p , p , m ) { displaystyle W ( mathbf {I} _ {p, p}, m)} Kvadratik shakl m d T M − 1 d { displaystyle md ^ {T} M ^ {- 1} d} T 2 ( p , m ) { displaystyle T ^ {2} (p, m)} p { displaystyle p} m { displaystyle m} [3]
Agar tasodifiy o'zgaruvchi bo'lsa X Hotellingnikiga ega T - kvadrat taqsimot, X ∼ T p , m 2 { displaystyle X sim T_ {p, m} ^ {2}} [1]
m − p + 1 p m X ∼ F p , m − p + 1 { displaystyle { frac {m-p + 1} {pm}} X sim F_ {p, m-p + 1}} qayerda F p , m − p + 1 { displaystyle F_ {p, m-p + 1}} F - tarqatishp va m − p + 1 .
T-kvadratik statistikani yozish
Ruxsat bering Σ ^ { displaystyle { hat { mathbf { Sigma}}}} namunaviy kovaryans :
Σ ^ = 1 n − 1 ∑ men = 1 n ( x men − x ¯ ) ( x men − x ¯ ) ′ { displaystyle { hat { mathbf { Sigma}}} = { frac {1} {n-1}} sum _ {i = 1} ^ {n} ( mathbf {x} _ {i} - { overline { mathbf {x}}}) ( mathbf {x} _ {i} - { overline { mathbf {x}}}) '} bu erda biz belgilaymiz ko'chirish tomonidan apostrof . Buni ko'rsatish mumkin Σ ^ { displaystyle { hat { mathbf { Sigma}}}} ijobiy (yarim) aniq matritsa va ( n − 1 ) Σ ^ { displaystyle (n-1) { hat { mathbf { Sigma}}}} p - o'zgaruvchan Istaklarni tarqatish bilan n −1 daraja erkinlik.[4] Σ ^ x ¯ = Σ ^ / n { displaystyle { hat { mathbf { Sigma}}} _ { overline { mathbf {x}}} = { hat { mathbf { Sigma}}} / n} [tushuntirish kerak
The Hotelling t - kvadrat statistikasi keyin quyidagicha aniqlanadi:[5]
t 2 = ( x ¯ − m ) ′ Σ ^ x ¯ − 1 ( x ¯ − m ) , { displaystyle t ^ {2} = ({ overline { mathbf {x}}} - { boldsymbol { mu}}) '{ hat { mathbf { Sigma}}} _ { overline { mathbf {x}}} ^ {- 1} ({ overline { mathbf {x}}} - { boldsymbol { mathbf { mu}}}),} bilan mutanosib bo'lgan masofa namuna o'rtacha va o'rtasida m { displaystyle { boldsymbol { mu}}} x ¯ ≅ m { displaystyle { overline { mathbf {x}}} cong { boldsymbol { mu}}}
Dan tarqatish ,
t 2 ∼ T p , n − 1 2 = p ( n − 1 ) n − p F p , n − p , { displaystyle t ^ {2} sim T_ {p, n-1} ^ {2} = { frac {p (n-1)} {n-p}} F_ {p, n-p},} qayerda F p , n − p { displaystyle F_ {p, n-p}} F - tarqatishp va n − p . A hisoblash uchun p - qiymatp o'zgaruvchan bu erda), ning taqsimlanishiga e'tibor bering t 2 { displaystyle t ^ {2}}
n − p p ( n − 1 ) t 2 ∼ F p , n − p . { displaystyle { frac {n-p} {p (n-1)}} t ^ {2} sim F_ {p, n-p}.} So'ngra, chap tomonning miqdorini baholash uchun foydalaning p -dan keladigan namunaga mos keladigan qiymat F - tarqatish. A ishonch mintaqasi shunga o'xshash mantiq yordamida ham aniqlanishi mumkin.
Motivatsiya Ruxsat bering N p ( m , Σ ) { displaystyle { mathcal {N}} _ {p} ({ boldsymbol { mu}}, { mathbf { Sigma}})} p - normal taqsimotni o'zgartirishManzil m { displaystyle { boldsymbol { mu}}} kovaryans Σ { displaystyle { mathbf { Sigma}}}
x 1 , … , x n ∼ N p ( m , Σ ) { displaystyle { mathbf {x}} _ {1}, dots, { mathbf {x}} _ {n} sim { mathcal {N}} _ {p} ({ boldsymbol { mu} }, { mathbf { Sigma}})} bo'lishi n mustaqil bir xil taqsimlangan (iid) tasodifiy o'zgaruvchilar sifatida ifodalanishi mumkin p × 1 { displaystyle p times 1}
x ¯ = x 1 + ⋯ + x n n { displaystyle { overline { mathbf {x}}} = { frac { mathbf {x} _ {1} + cdots + mathbf {x} _ {n}} {n}}} bo'lish namuna o'rtacha kovaryans bilan Σ x ¯ = Σ / n { displaystyle { mathbf { Sigma}} _ { bar { mathbf {x}}} = { mathbf { Sigma}} / n}
( x ¯ − m ) ′ Σ x ¯ − 1 ( x ¯ − m ) ∼ χ p 2 , { displaystyle ({ bar { mathbf {x}}} - { boldsymbol { mu}}) '{ mathbf { Sigma}} _ { bar { mathbf {x}}} ^ {- 1 } ({ bar { mathbf {x}}} - { boldsymbol { mathbf { mu}}}) sim chi _ {p} ^ {2},} qayerda χ p 2 { displaystyle chi _ {p} ^ {2}} kvadratchalar bo'yicha taqsimlash bilan p erkinlik darajasi.[6]
Isbot —
Buni ko'rsatish uchun haqiqatdan foydalaning x ¯ ∼ N p ( m , Σ / n ) { displaystyle { overline { mathbf {x}}} sim { mathcal {N}} _ {p} ({ boldsymbol { mu}}, { mathbf { Sigma}} / n)} xarakterli funktsiya tasodifiy o'zgaruvchining y = ( x ¯ − m ) ′ Σ x ¯ − 1 ( x ¯ − m ) = ( x ¯ − m ) ′ ( Σ / n ) − 1 ( x ¯ − m ) { displaystyle mathbf {y} = ({ bar { mathbf {x}}} - { boldsymbol { mu}}) '{ mathbf { Sigma}} _ { bar { mathbf {x} }} ^ {- 1} ({ bar { mathbf {x}}} - { boldsymbol { mathbf { mu}}}) = ({ bar { mathbf {x}}} - { boldsymbol { mu}}) '({ mathbf { Sigma}} / n) ^ {- 1} ({ bar { mathbf {x}}} - { boldsymbol { mathbf { mu}}}) } | ⋅ | { displaystyle | cdot |} aniqlovchi kabi, argumentning | Σ | { displaystyle | { boldsymbol { Sigma}} |}
Xarakterli funktsiya ta'rifi bo'yicha biz quyidagilarga egamiz:[7]
φ y ( θ ) = E e men θ y , = E e men θ ( x ¯ − m ) ′ ( Σ / n ) − 1 ( x ¯ − m ) = ∫ e men θ ( x ¯ − m ) ′ n Σ − 1 ( x ¯ − m ) ( 2 π ) − p / 2 | Σ / n | − 1 / 2 e − ( 1 / 2 ) ( x ¯ − m ) ′ n Σ − 1 ( x ¯ − m ) d x 1 ⋯ d x p { displaystyle { begin {aligned} varphi _ { mathbf {y}} ( theta) & = operator nomi {E} e ^ {i theta mathbf {y}}, [5pt] & = operatorname {E} e ^ {i theta ({ overline { mathbf {x}}} - { boldsymbol { mu}}) '({ mathbf { Sigma}} / n) ^ {- 1 } ({ overline { mathbf {x}}} - { boldsymbol { mathbf { mu}}})} [5pt] & = int e ^ {i theta ({ overline { mathbf) {x}}} - { boldsymbol { mu}}) 'n { mathbf { Sigma}} ^ {- 1} ({ overline { mathbf {x}}} - { boldsymbol { mathbf { mu}}})} (2 pi) ^ {- p / 2} | { boldsymbol { Sigma}} / n | ^ {- 1/2} , e ^ {- (1/2) ( { overline { mathbf {x}}} - { boldsymbol { mu}}) 'n { boldsymbol { Sigma}} ^ {- 1} ({ overline { mathbf {x}}} - { boldsymbol { mu}})} , dx_ {1} cdots dx_ {p} end {aligned}}} Integral ichida ikkita eksponentlar mavjud, shuning uchun eksponentlarni ko'paytirib, ko'rsatkichlarni birlashtiramiz va quyidagilarni olamiz:
= ∫ ( 2 π ) − p / 2 | Σ / n | − 1 / 2 e − ( 1 / 2 ) ( x ¯ − m ) ′ n ( Σ − 1 − 2 men θ Σ − 1 ) ( x ¯ − m ) d x 1 ⋯ d x p { displaystyle { begin {aligned} & = int (2 pi) ^ {- p / 2} | { boldsymbol { Sigma}} / n | ^ {- 1/2} , e ^ {- (1/2) ({ overline { mathbf {x}}} - { boldsymbol { mu}}) 'n ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ({ overline { mathbf {x}}} - { boldsymbol { mu}})} , dx_ {1} cdots dx_ {p} end {hizalanmış }}} Endi muddatni oling | Σ / n | − 1 / 2 { displaystyle | { boldsymbol { Sigma}} / n | ^ {- 1/2}} Men = | ( Σ − 1 − 2 men θ Σ − 1 ) − 1 / n | 1 / 2 ⋅ | ( Σ − 1 − 2 men θ Σ − 1 ) − 1 / n | − 1 / 2 { displaystyle I = | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} / n | ^ {1 / 2} ; cdot ; | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} / n | ^ {-1/2}}
= | ( Σ − 1 − 2 men θ Σ − 1 ) − 1 / n | 1 / 2 | Σ / n | − 1 / 2 ∫ ( 2 π ) − p / 2 | ( Σ − 1 − 2 men θ Σ − 1 ) − 1 / n | − 1 / 2 e − ( 1 / 2 ) n ( x ¯ − m ) ′ ( Σ − 1 − 2 men θ Σ − 1 ) ( x ¯ − m ) d x 1 ⋯ d x p { displaystyle { begin {aligned} & = | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} / n | ^ {1/2} | { boldsymbol { Sigma}} / n | ^ {- 1/2} int (2 pi) ^ {- p / 2} | ({ boldsymbol { Sigma} } ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} / n | ^ {- 1/2} , e ^ {- (1/2) n ({ overline { mathbf {x}}} - { boldsymbol { mu}}) '({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ({ overline { mathbf {x}}} - { boldsymbol { mu}})} , dx_ {1} cdots dx_ {p} end {aligned}}} Ammo integral ichidagi atama aniq $ a $ ning zichlik funktsiyasidir ko'p o'zgaruvchan normal taqsimot kovaryans matritsasi bilan ( Σ − 1 − 2 men θ Σ − 1 ) − 1 / n = [ n ( Σ − 1 − 2 men θ Σ − 1 ) ] − 1 { displaystyle ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} / n = left [n ({) boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) right] ^ {- 1}} m { displaystyle mu} x 1 , … , x p { displaystyle x_ {1}, dots, x_ {p}} 1 { displaystyle 1} ehtimollik aksiomalari .[tushuntirish kerak Shunday qilib:
= | ( Σ − 1 − 2 men θ Σ − 1 ) − 1 ⋅ 1 n | 1 / 2 | Σ / n | − 1 / 2 = | ( Σ − 1 − 2 men θ Σ − 1 ) − 1 ⋅ 1 n ⋅ n ⋅ Σ − 1 | 1 / 2 = | [ ( Σ − 1 − 2 men θ Σ − 1 ) Σ ] − 1 | 1 / 2 = | Men p − 2 men θ Men p | − 1 / 2 { displaystyle { begin {aligned} & = left | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1 } cdot { frac {1} {n}} right | ^ {1/2} | { boldsymbol { Sigma}} / n | ^ {- 1/2} & = left | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} cdot { frac {1} { bekor {n}} } cdot { bekor qilish {n}} cdot { boldsymbol { Sigma}} ^ {- 1} o'ng | ^ {1/2} & = chap | chap [({ bekor qilish {{ boldsymbol { Sigma}} ^ {- 1}}} - 2i theta { bekor qilish {{ boldsymbol { Sigma}} ^ {- 1}}}) { bekor qilish { boldsymbol { Sigma}}} right] ^ {- 1} right | ^ {1/2} & = | mathbf {I} _ {p} -2i theta mathbf {I} _ {p} | ^ {- 1 / 2} end {hizalangan}}} qayerda Men p { displaystyle I_ {p}} p { displaystyle p}
= ( 1 − 2 men θ ) − p / 2 { displaystyle { begin {aligned} & = (1-2i theta) ^ {- p / 2} end {aligned}}} bu uchun xarakterli funktsiya xi-kvadrat taqsimot bilan p { displaystyle p} ◼ { displaystyle ; ; ; blacksquare}
Ikki namunali statistika
Agar x 1 , … , x n x ∼ N p ( m , V ) { displaystyle { mathbf {x}} _ {1}, dots, { mathbf {x}} _ {n_ {x}} sim N_ {p} ({ boldsymbol { mu}}, { mathbf {V}})} y 1 , … , y n y ∼ N p ( m , V ) { displaystyle { mathbf {y}} _ {1}, dots, { mathbf {y}} _ {n_ {y}} sim N_ {p} ({ boldsymbol { mu}}, { mathbf {V}})} mustaqil ravishda ikkitadan chizilgan mustaqil ko'p o'zgaruvchan normal taqsimotlar bir xil o'rtacha va kovaryans bilan va biz aniqlaymiz
x ¯ = 1 n x ∑ men = 1 n x x men y ¯ = 1 n y ∑ men = 1 n y y men { displaystyle { overline { mathbf {x}}} = { frac {1} {n_ {x}}} sum _ {i = 1} ^ {n_ {x}} mathbf {x} _ { i} qquad { overline { mathbf {y}}} = { frac {1} {n_ {y}}} sum _ {i = 1} ^ {n_ {y}} mathbf {y} _ {i}} namuna sifatida va
Σ ^ x = 1 n x − 1 ∑ men = 1 n x ( x men − x ¯ ) ( x men − x ¯ ) ′ { displaystyle { hat { mathbf { Sigma}}} _ { mathbf {x}} = { frac {1} {n_ {x} -1}} sum _ {i = 1} ^ {n_ {x}} ( mathbf {x} _ {i} - { overline { mathbf {x}}}) ( mathbf {x} _ {i} - { overline { mathbf {x}}}) '} Σ ^ y = 1 n y − 1 ∑ men = 1 n y ( y men − y ¯ ) ( y men − y ¯ ) ′ { displaystyle { hat { mathbf { Sigma}}} _ { mathbf {y}} = { frac {1} {n_ {y} -1}} sum _ {i = 1} ^ {n_ {y}} ( mathbf {y} _ {i} - { overline { mathbf {y}}}) ( mathbf {y} _ {i} - { overline { mathbf {y}}}) '} tegishli namunaviy kovaryans matritsalari sifatida. Keyin
Σ ^ = ( n x − 1 ) Σ ^ x + ( n y − 1 ) Σ ^ y n x + n y − 2 { displaystyle { hat { mathbf { Sigma}}} = { frac {(n_ {x} -1) { hat { mathbf { Sigma}}} _ { mathbf {x}} + ( n_ {y} -1) { hat { mathbf { Sigma}}} _ { mathbf {y}}} {n_ {x} + n_ {y} -2}}} xolisdir birlashtirilgan kovaryans matritsasi smeta (kengaytmasi birlashtirilgan dispersiya ).
Va nihoyat Hotellingning ikkita namunasi t - kvadrat statistikasi bu
t 2 = n x n y n x + n y ( x ¯ − y ¯ ) ′ Σ ^ − 1 ( x ¯ − y ¯ ) ∼ T 2 ( p , n x + n y − 2 ) { displaystyle t ^ {2} = { frac {n_ {x} n_ {y}} {n_ {x} + n_ {y}}} ({ overline { mathbf {x}}} - { overline { mathbf {y}}}) '{ hat { mathbf { Sigma}}} ^ {- 1} ({ overline { mathbf {x}}} - { overline { mathbf {y}} }) sim T ^ {2} (p, n_ {x} + n_ {y} -2)} Tegishli tushunchalar Bu F-tarqatish bilan bog'liq bo'lishi mumkin[4]
n x + n y − p − 1 ( n x + n y − 2 ) p t 2 ∼ F ( p , n x + n y − 1 − p ) . { displaystyle { frac {n_ {x} + n_ {y} -p-1} {(n_ {x} + n_ {y} -2) p}} t ^ {2} sim F (p, n_) {x} + n_ {y} -1-p).} Ushbu statistikaning null bo'lmagan taqsimoti quyidagicha markazdan tashqari F-taqsimot (a nisbati markaziy bo'lmagan kvadratchalar tasodifiy o'zgaruvchi va mustaqil markaziy Kvadratchalar tasodifiy o'zgaruvchi)
n x + n y − p − 1 ( n x + n y − 2 ) p t 2 ∼ F ( p , n x + n y − 1 − p ; δ ) , { displaystyle { frac {n_ {x} + n_ {y} -p-1} {(n_ {x} + n_ {y} -2) p}} t ^ {2} sim F (p, n_) {x} + n_ {y} -1-p; delta),} bilan
δ = n x n y n x + n y ν ′ V − 1 ν , { displaystyle delta = { frac {n_ {x} n_ {y}} {n_ {x} + n_ {y}}} { boldsymbol { nu}} ' mathbf {V} ^ {- 1} { boldsymbol { nu}},} qayerda ν = x ¯ − y ¯ { displaystyle { boldsymbol { nu}} = mathbf {{ overline {x}} - { overline {y}}}}
Ikki o'zgaruvchan holatda, formulalar qanday qilib o'zaro bog'liqligini baholashga imkon beradigan tarzda soddalashtiradi, r { displaystyle rho} t 2 { displaystyle t ^ {2}}
d 1 = x ¯ 1 − y ¯ 1 , d 2 = x ¯ 2 − y ¯ 2 { displaystyle d_ {1} = { overline {x}} _ {1} - { overline {y}} _ {1}, qquad d_ {2} = { overline {x}} _ {2} - { overline {y}} _ {2}} va
s 1 = V 11 s 2 = V 22 r = V 12 / ( s 1 s 2 ) = V 21 / ( s 1 s 2 ) { displaystyle s_ {1} = { sqrt {W_ {11}}} qquad s_ {2} = { sqrt {W_ {22}}} qquad rho = W_ {12} / (s_ {1} s_ {2}) = W_ {21} / (s_ {1} s_ {2})} keyin
t 2 = n x n y ( n x + n y ) ( 1 − r 2 ) [ ( d 1 s 1 ) 2 + ( d 2 s 2 ) 2 − 2 r ( d 1 s 1 ) ( d 2 s 2 ) ] { displaystyle t ^ {2} = { frac {n_ {x} n_ {y}} {(n_ {x} + n_ {y}) (1-r ^ {2})}} chap [ chap ({ frac {d_ {1}} {s_ {1}}} o'ng) ^ {2} + chap ({ frac {d_ {2}} {s_ {2}}} o'ng) ^ {2 } -2 rho chap ({ frac {d_ {1}} {s_ {1}}} o'ng) chap ({ frac {d_ {2}} {s_ {2}}} o'ng) o'ngda]} Shunday qilib, agar vektorning ikki qatoridagi farqlar ( x ¯ − y ¯ ) { displaystyle ({ overline { mathbf {x}}} - { overline { mathbf {y}}})} t 2 { displaystyle t ^ {2}} r { displaystyle rho} t 2 { displaystyle t ^ {2}} r { displaystyle rho}
Bitta o'zgaruvchan maxsus holatni topish mumkin Welchning t-testi .
Adabiyotda Hotellingning ikkita namunali testiga qaraganda ancha kuchli va kuchli testlar taklif qilingan, masalan, o'zgaruvchilar soni mavzular soni bilan taqqoslanadigan yoki hatto undan ham kattaroq bo'lganda qo'llanilishi mumkin bo'lgan intervalgacha masofaga asoslangan testlarga qarang.[8] [9]
Shuningdek qarang
Adabiyotlar
^ a b v Hotelling, H. (1931). "Talabalar koeffitsientini umumlashtirish" . Matematik statistika yilnomalari 2 (3): 360–378. doi :10.1214 / aoms / 1177732979 ^ Jonson, RA .; Wichern, D.W. (2002). Amaliy ko'p o'zgaruvchan statistik tahlil . 5 . Prentice zali. ^ Erik Vaytshteyn, MathWorld ^ a b Mardiya, K. V .; Kent, J. T .; Bibbi, J. M. (1979). Ko'p o'zgaruvchan tahlil . Akademik matbuot. ISBN 978-0-12-471250-8 ^ "6.5.4.3. Hotellingning T kvadrat " .^ 4.2-bobning oxiri Jonson, R.A. & Wichern, D.W. (2002) ^ Billingsley, P. (1995). "26. Xarakterli funktsiyalar". Ehtimollik va o'lchov (3-nashr). Vili. ISBN 978-0-471-00710-4 ^ Marozzi, M. (2016). "Magnit-rezonansli tomografiya qilish uchun interpekt masofalarga asoslangan ko'p o'zgaruvchan testlar". Tibbiy tadqiqotlarda statistik usullar . 25 (6): 2593–2610. doi :10.1177/0962280214529104 . PMID 24740998 . ^ Marozzi, M. (2015). "Katta o'lchovli past namunali o'lchovli vaziyatni nazorat qilish tadqiqotlari uchun ko'p o'lchovli ko'p o'lchovli sinovlar". Tibbiyotdagi statistika . 34 (9): 1511–1526. doi :10.1002 / sim.6418 . PMID 25630579 . Tashqi havolalar
Diskret o'zgaruvchan Diskret o'zgaruvchan Doimiy o'zgaruvchan Doimiy o'zgaruvchan Doimiy o'zgaruvchan Doimiy o'zgaruvchan Aralashtirilgan uzluksiz diskret bir o'zgaruvchidir Ko'p o'zgaruvchan (qo'shma) Yo'naltirilgan Degeneratsiya va yakka Oilalar
![{ displaystyle { begin {aligned} varphi _ { mathbf {y}} ( theta) & = operator nomi {E} e ^ {i theta mathbf {y}}, [5pt] & = operatorname {E} e ^ {i theta ({ overline { mathbf {x}}} - { boldsymbol { mu}}) '({ mathbf { Sigma}} / n) ^ {- 1 } ({ overline { mathbf {x}}} - { boldsymbol { mathbf { mu}}})} [5pt] & = int e ^ {i theta ({ overline { mathbf) {x}}} - { boldsymbol { mu}}) 'n { mathbf { Sigma}} ^ {- 1} ({ overline { mathbf {x}}} - { boldsymbol { mathbf { mu}}})} (2 pi) ^ {- p / 2} | { boldsymbol { Sigma}} / n | ^ {- 1/2} , e ^ {- (1/2) ( { overline { mathbf {x}}} - { boldsymbol { mu}}) 'n { boldsymbol { Sigma}} ^ {- 1} ({ overline { mathbf {x}}} - { boldsymbol { mu}})} , dx_ {1} cdots dx_ {p} end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f7443b5a91e8899f181974528a7d34bf9e047a)
![{ displaystyle ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} / n = left [n ({) boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) right] ^ {- 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e30acda26292ba3fcf5d6d302141d34fcebce5)
![{ displaystyle { begin {aligned} & = left | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1 } cdot { frac {1} {n}} right | ^ {1/2} | { boldsymbol { Sigma}} / n | ^ {- 1/2} & = left | ({ boldsymbol { Sigma}} ^ {- 1} -2i theta { boldsymbol { Sigma}} ^ {- 1}) ^ {- 1} cdot { frac {1} { bekor {n}} } cdot { bekor qilish {n}} cdot { boldsymbol { Sigma}} ^ {- 1} o'ng | ^ {1/2} & = chap | chap [({ bekor qilish {{ boldsymbol { Sigma}} ^ {- 1}}} - 2i theta { bekor qilish {{ boldsymbol { Sigma}} ^ {- 1}}}) { bekor qilish { boldsymbol { Sigma}}} right] ^ {- 1} right | ^ {1/2} & = | mathbf {I} _ {p} -2i theta mathbf {I} _ {p} | ^ {- 1 / 2} end {hizalangan}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b99da9756cc4c6b0312d41367697e0aa53eaca)
![{ displaystyle t ^ {2} = { frac {n_ {x} n_ {y}} {(n_ {x} + n_ {y}) (1-r ^ {2})}} chap [ chap ({ frac {d_ {1}} {s_ {1}}} o'ng) ^ {2} + chap ({ frac {d_ {2}} {s_ {2}}} o'ng) ^ {2 } -2 rho chap ({ frac {d_ {1}} {s_ {1}}} o'ng) chap ({ frac {d_ {2}} {s_ {2}}} o'ng) o'ngda]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdef78058fb5071adade5aaa4362df16e1b3747b)
