Γ2
X 1
X =
X d
E (X ) =
Σ = C ov( ov(X, X ) =
E ( E (X 1 ) E ( E (X d )
V ar (X 1 ) Cov (X 1 , X 2 ) Cov (X 2 , X 1 ) V ar (X 2 )
··· ···
C ov (X 1 , X d ) C ov (X 2 , X d )
Cov (X d , X 1 ) Cov (X d , X 2 )
···
V ar (X d )
β
q
≤d
A
B
q
×d
α α
Y = Σ−1/2 (X E (Y ) Y ) = 0
− µ)
Cov( Cov (Y, Y ) Y ) = I d Σ−1/2
A
A = (v1 , . . . , vd ) v1 , . . . , vd
Cov( Cov (α + AX,β + AX,β + BY ) BY ) = ACov( ACov (X, Y ) Y )B
E (α + AX ) = α + AE (X ) α
λ1
0
0
λd
v1
vd
A λ1 , . . . , λd A 2 x = x Ax
k Ak = (v1 , . . . , vd )
X 1 , . . . , Xn
v1
λk1
0
0
λkd
vd
∈ N d(µ, Σ) ¯= 1 X n
n
X i
i=1
∈ N d
1 µ, Σ n µ
¯ X ¯ n X
¯ µ Σ−1 X
− − ∈
1
∈ µ
µ
χ2d
−α
R
d
¯ : n X
−1
¯ X
− − µ Σ
µ <
d = 2
λ j
χ2d,α /n
j
∈{
λ1 , . . . , λd
}
d = 3
Σ
d=2 ¯ X +
v1
Σ 1, . . . , d
χ2d,α
¯ X
d R
∈R
v2
χ2d,α /n
λ1 v1 cos(θ) + Σ
λ1
λ2
λ2 v2
(θ)
θ
∈ [0, 2π)
T
X 1 , . . . , Xn ¯= 1 X n
n
X i
i=1
∈ N d
∈ N d(µ, Σ)
1 µ, Σ n
1 nS = n
n
X i
i=1
− X ¯
X i
− X ¯ ∈ W d(Σ, n − 1)
Γ2 (n
− 1) n
−d d
¯ µ S −1 X
¯ X
− − ∈ µ
¯ µ S −1 X
¯ X
Γ2 (d, n F
− − ∈
Σ=
µ
2.5 1 1 2
X 1 , . . . , Xn
− 1)
F d,(n−d)
∈ N d(µ, Σ) µ
Σ
L(x,µ, Σ) = (2π)−nd/2 Σ −n/2 exp
| |
n
− 1 2
i=1
(xi
− µ) Σ−1 (xi − µ)
n
n
(xi
i=1
−1
− µ) Σ
(xi
− µ)
− x¯) Σ−1 (xi − x¯) + (¯x − µ) Σ−1 (¯x − µ) i=1 +2(¯ x − µ) Σ−1 (xi − x ¯) n (xi − x ¯) Σ−1 (xi − x¯) + n (¯ x − µ) Σ−1 (¯ x − µ)
=
(xi
=
i=1
log L(x,µ, Σ) = c = c c=
−
n log Σ 2
| |−
1 2
− n2 log |Σ| − 12
− nd2 log(2π)
n
(xi
− µ) Σ−1 (xi − µ)
(xi
− x¯) Σ−1 (xi − x¯) − n2 (¯x − µ) Σ−1 (¯x − µ)
i=1 n i=1
Σ−1
Σ µ) Σ−1 (¯ x
− µ) > 0
µ = x ¯ µ ˆ =x ¯
sup log L(x,µ, Σ) = c µ
−
n log Σ 2
| |−
1 2
Σ
n
(xi
i=1
− x¯) Σ−1 (xi − x¯) Σ
sup log L(x,µ, Σ) = c µ
= c = c = c
− n2 log |Σ| − 12 − n2 log |Σ| − 12 − n2 log |Σ| − 12 −
n log Σ + 2
| |
n
− − (xi
x ¯) Σ−1 (xi
− x¯)
(xi
x ¯) Σ−1 (xi
− x¯)
i=1
n
i=1 n
Σ−1 (xi
i=1
− x¯) (xi − x¯)
Σ−1 S
Σ
A f (Σ) = log Σ + traza Σ−1 A ,
| |
Σ
Σ=A
(¯ x
−
¯ X sup sup log L(x,µ, Σ) = c Σ
µ
− n2
S −1 S
log S +
||
X 1 , . . . , Xn
=c
− n2 (log |S | + d)
∈ N d(µ, Σ)
Σ µ µ
H 0 : µ = µ0 H 0 : µ = µ0 µ0 , Σ) −2log λ(X ) = −2log supL(X,L(X,µ, Σ) µ
H 0 : µ = µ0
log L(x, µ0 , Σ) = c
−
n log Σ 2
| |−
1 2
sup log L(x,µ, Σ) = c µ
n
i=1
−
(xi
− x¯) Σ−1 (xi − x¯) − n2 (¯x − µ0) Σ−1 (¯x − µ0)
n log Σ 2
| |−
1 2
n
− x¯) Σ−1 (xi − x¯)
¯ X
− µ0 Σ−1
i=1
µ0 , Σ) =n −2log λ(X ) = −2log supL(X,L(X,µ, Σ) µ
(xi
¯ X
− µ0
S
H 0 : µ = µ0 ¯ X
∈ N d (µ0, Σ/n)
¯ µ0 Σ−1 X
¯ n X
− − ∈ − − µ0
χ2d
H 0 : µ = µ0
¯ n X χ2d,α
1
¯ µ0 Σ−1 X
µ0 > χ2d,α
χ2d
−α
X 1 , . . . , Xn
∈ N d(µ, Σ) µ
Σ H 0 : µ = µ0 supΣ L(X, µ0 , Σ) −2log λ(X ) = −2log sup L(X,µ, Σ) µ,Σ
Σ
¯ S X
sup sup log L(x,µ, Σ) = c µ
Σ
log L(x, µ0 , Σ) = c
ˆµ = Σ 0
1 n
n i=1 (xi
−
− n2
||
n log Σ 2
| |−
1 2
= c
− n2 log |Σ| − 12
= c
n log Σ + 2
−
S −1 S
log S +
| |
− µ0) (xi − µ0) µ0
=c
− n2 (log |S | + d)
n
− −
µ0 ) Σ−1 (xi
(xi
i=1
− µ0 )
n
Σ−1 (xi
i=1
ˆµ Σ−1 Σ
0
µ0 x ¯
µ0 ) (xi
− µ0)
ˆµ Σ
0
sup log L(x, µ0 , Σ) = c Σ
− n2
| | 0
supΣ L(X, µ0 , Σ) −2log λ(X ) = −2log sup =n L(X,µ, Σ) µ,Σ
ˆµ Σ
0
= =
1 n 1 n
(X i
X i
¯ + X ¯ X
¯ X i X
i=1
−2log λ(X )
= n log S + rr
I + S rr
| | − log |S | 0
− µ0
¯ X
I + S −1 rr
= n log S
=
(b)
¯ X
− µ0
X i
− X ¯
log S = n log I + S −1 rr
(c)
1 + λ1 , . . . , 1 + λd
S −1 rr d
log S
S −1 rr
− µ0 + 2
| | − | | | |· − ||
λ1 , . . . , λd I + S −1 rr (a)
ˆµ log Σ
µ0 )
µ0 ) (X i
i=1 n
−
−1
n
− − − −
= S + rr ¯ µ0 r = X
ˆµ + d log Σ
(1 + λ j ) = 1 + λ1 = 1 + traza S −1 rr = 1 + traza r S −1 r = 1 + r S −1 r
j=1
1 + r S −1 r
−2log λ(X ) = n log
¯ r S −1 r = X
n
−d d
n
−d d
¯ X
¯ X
− µ0 S −1
− µ0 S −1
¯ X
¯ X
¯ X
− µ0 ∈ F d,n−d
H 0 : µ = µ0
− µ0 S −1
− µ0
− µ0
> f d,n−d,α
Γ2
∈ µ
d
:
R
n
−d d
¯ X
−1
¯ X
− − µ S
µ < f d,n−d,α
¯ X
d
R
S
λ j
λ1 , . . . , λd
n
d
− d f d,n−d,α
∈ {1, . . . , d}
S
l µ = l1 µ1 + l X 1 , . . . , l X n ¯ l X
j
·· · + ldµd
∈ N (lµ, lΣl) l X 1 , . . . , l X n
l S c l
√ lS l √ lS l c ¯ − tn−1,α/2 √ , l X + ¯ tn−1,α/2 √ c l X n n
1
tn−1,α/2 l
− α/2
T
n
l µ l =
(1, 0, . . . , 0) p
l tn−1,α/2 P
√ − √
¯ l µ n l X l S c l
∀l ∈ R p tn−1,α/2
c
−1
=1
−α
1
−α
B
d
×d
r
∈ Rd
(x r)2 max = r B −1 r x∈R \{0} x Bx d
x = cB −1 r
¯ µ n l X max l S c l l∈R
c
2
∈ R\{0}
− − − ∈
d
¯ = n X
¯ µ S c−1 X
µ
− −
Γ2 (d, n
− 1)
c
√ √
− −
¯ l X
d(n 1) l S c l ¯ f d,n−d,α , l X + n d n
−
¯ l X
−
n
d
−d
¯ f d,n−d,α l Sl , l X +
√ √
d(n 1) l S c l f d,n−d,α n d n
n
d
− d f d,n−d,αl Sl
c Γ2
T
d(n−1) n−d f d,n−d,0 05
n
tn−1,0 025
d=4
d = 10
∞
C 1 , . . . , Cm P (C i
∀i)
= 1
− P (C i m ≥ 1 − P (C i
i=1
i) ) =1
− (α1 + ·· · + αm)
1 α1 , . . . , 1 αm
−
−
α1 =
·· ·
1 α = αm = α/m
α1 , . . . , αm
−
1
− α = 095 =
α1 +
· · ·+αm = α
tn−1,α/(2m)
d(n−1) n−d f d,n−d,α
m=d n
∞
µ
H 0 : µ = µ0 µ
X 1 , . . . , Xn H a
¯ X
µ ˆ Σ Σ
S +
rr
S
H 0
¯ r = X
H a
H 0
−2log λ(X ) = nrΣ−1r −2log λ(X ) = n log
1 + r S −1 r
N d (µ, Σ)
Σ
Σ
− µˆ H a
H 0
Σ H 0 : Bµ = b B
q
×d
q
b
H 0
l(x,µ, Σ) = log L(x,µ, Σ) = c
−
n log Σ 2
| |−
1 2
n
(xi
i=1
µ ˆ
− x¯) Σ−1 (xi − x¯) − n2 (¯x − µ) Σ−1 (¯x − µ)
l(x,µ, Σ) Bµ = b
l+ = l
− nλ(Bµ − b)
λ ∂l + = n (¯ x ∂µ x ¯
− µ) Σ−1 − nλB = 0 − µ = ΣBλ Bµ = b
λ
µ
B
− Bµ = Bx¯ − b = BΣB λ = (BΣB )−1 (B x ¯ − b)
Bx ¯
¯ µ ˆ = X
¯ r = X
− ΣB
− µˆ = ΣB
−2log λ(X ) = n
−1
¯ B X
−b
−1
¯ B X
b
BΣB
−1
¯ B X
BΣB
− − − ¯ B X
H 0 BX 1 , . . . , B Xn b
BΣB
λ
b
∈ N q (b, BΣB)
b
q χ2q
µ ¯ µ ˆ = X
¯ r = X
− SB
BSB
− µˆ = SB
− 1)rS −1r = (n − 1) Γ2 (q, n − 1)
(n
−1
BSB
¯ B X
−1
−b
¯ B X
− − ¯ B X
b
BSB
b
−1
B=
H 0 : Bµ = 0
1
−1
0
1
0
−1
1
¯ B X
−b
µ = (µ1 , . . . , µd )
d
H 0
0
·· ·
· ··
0 0 1
0
− d
B
µ H 0 : Σ = Σ0
supµ L (X,µ, Σ0 ) −2log λ(X ) = −2log sup L (X,µ, Σ) µ,Σ
sup log L (X,µ, Σ0 ) = c µ
− n2
sup sup log L(x,µ, Σ) = c Σ
−2log λ(X )
|
µ
1 Σ− 0 S
= n log Σ0 + d
= n
j=1
|
λ j
1 Σ− 0 S
log Σ0 +
| |
− n2 (log |S | + d) 1 Σ− 0 S
1 log Σ− 0 S
− | |− − − − − − − − log S
d =n
d
d
− log
λ j
d
= n da
log g d
pd = nd (a
log g
1)
j=1
1 Σ− 0 S a
λ1 , . . . , λd g
−2log λ(X ) = nd (a − log g − 1) ∼ χ2m m = 12 d(d + 1)
(1
− α)
χ2m
Σ=
χ2m,α
2.5 1 1 2
−2log λ(X ) = nd (a − log g − 1)
α
1ˆ Σ− 0 Σµ
a
g S ˆµ = 1 Σ n
n
(X i
i=1
− µ) (X i − µ)
χ2m
m = 12 d(d + 1)
X 1 , . . . , Xn
N d (µ, Σ) S
ˆ Σ
H a
H 0 ¯ X
µ H 0
H a
−2log λ(X ) = nd (a − log g − 1) ˆ −1 S Σ
a
g
H 0 : Σ = kΣ0 , k
∈ (0, +∞)
µ
µ H 0 : Σ = kΣ0
k
∈ (0, +∞)
Σ0
ˆ 0 ˆ = kΣ Σ µ sup log L (X,µ,kΣ0 ) = c µ
−
n log kΣ0 + 2
| |
−1
(kΣ0 )
S
= c
−
= c
−
n 2 n 2
| | | d
log k Σ0
+
d log k + log Σ0 +
|
1 k
1 −1 Σ S k 0
1 Σ− 0 S
k ∂ supµ log L (X,µ,kΣ0 ) = ∂k 1 kˆ = d
−
n 2
1 d k
−
1 k2
1 Σ− 0 S
1 Σ− 0 S = a0
1 Σ− 0 S
a0 kˆ = a0
k
−2log λ(X ) = nd (a − log g − 1) a 1 − ˆ Σ S =
g
1 −1 a0 Σ0 S 1 Σ− 0 S
a = 1
−2log λ(X ) = nd
g = a
1 a0 g0
g0
g
− − 1
log
1 g0 a0
1
= nd log
m = 12 d(d + 1)
χ2m
H 0 : Σ = k I
·
Σ0 = I
−2log λ(X ) = nd log ag00 ∼ χ2m a0 g0 1 Σ− 0 S = S
m = 12 (d
− 1)(d + 2)
a0 g0
− 1 = 12 (d − 1)(d + 2)
H 0 : Σ12 = 0, d1
µ d2
d1 + d2 = d Σ=
Σ12 = 0 H 0 : Σ12 = 0
Σ11 Σ12 Σ21 Σ22
µ1
Σ11
µ2
µ = (µ1 , µ2 )
¯ 1 , X ¯ 2 ) = X ¯ µ ˆ = (X
ˆ= Σ
S 11 0 0 S 22
¯ X
ˆ −1
Σ
a=
ˆ −1 S = g = Σ d
S =
−1
S 11 0 0 S 22
S 11 S 12 S 21 S 22
=
S
−1 I S 11 S 12 −1 S 22 S 21 I
− = I
R = D−1/2 SD −1/2
λ(X )2/n = I
−
H 0 : Σ12 = 0
−1 −1 S 22 S 21 S 11 S 12
∈
Λ (d2 , d1 , n
M 11 = nS 11 M 22 = nS 22 M 22·1 = n S 22
−
−1 S 21 S 11 S 12
= M 22
∈ −
d2
∈
− 1 − d1)
d1 d2 d2
(Σ11 , n
− 1) (Σ22 , n − 1) (Σ22·1 , n − 1 − d1 )
(M 11 , M 22 )
H 0 : Σ12 = 0 M 22·1 M 22 M 22·1
−
−1 M 21 M 11 M 12
∈ ∈ ∈
−1 1 R− 22 R21 R11 R12
D
−2log λ(X ) = −nd log g = −n log I − S 22−1S 21S 11−1S 12
M 22·1
ˆ −1 S = 1 d = 1 Σ d
1 d
|S | = |S 22 − S 21S 11−1S 12| = I − S −1S 21S −1S 12 22 11 |S 11| · |S 22| |S 22|
Σ22
Σ22·1 = Σ22
(Σ22·1 , n 1 d1 ) d (Σ22·1 , d1 )
− −
2
⇒| =
|M 22·1| ∈ Λ (d2, d1, n − 1 − d1) M 22·1 + (M 22 − M 22·1 )|
λ(X )2/n = I
−
Σ12 = 0
−1 −1 S 22 S 21 S 11 S 12 =
|S 22 − S 21S 11−1S 12| = |M 22·1| |S 22| |M 22·1 + (M 22 − M 22·1)|
λ(X )
|I −
F
R11 = 1
λ(X )2/n = I R
(a)
1 R− 22 αα = 1
1
|
−
−1
− αR−221α = 1 − R12R22−1R21 = 1 − R2 ∈ Λ (d2, 1, n − 2)
− Λ(d, 1, m) = Λ(d, 1, m)
R2 1 R2
−
R2 /(1
H 0 :
d1 = 1 d2 = d d 1
α = R21
−
−1 −1 S 22 S 21 S 11 S 12
m
−
d F d,m−d+1 , d+1
∈ nd −− 1d F d−1,n−d
− R2 )
F H 0 : Σ
¯ X
µ
ˆ= Σ
S 12 S d2 ¯ X
S
µ
Σ
−2log λ(X ) = nd (a − log g − 1) = −n log |R| ˆ −1 S Σ
a
g R ˆ −1/2 S Σ ˆ −1/2 R = Σ
ˆ −1 S Σ
d a=1
−n log |R| ∼ χ2 d(d−1) 1 2
d + 12 d(d + 1)
X 11 , . . . , X1n
1
... X k1 , . . . , Xkn
k
∈
N d (µ1 , Σ1 )
∈
N d (µk , Σk )
...
k k
X 11 , . . . , X1n
1
X 21 , . . . , X2n
2
∈ ∈
N d (µ1 , Σ1 ) N d (µ2 , Σ2 )
Σ1 = Σ2
H 0 : µ1 = µ2
− (d + d) = 12 d(d − 1)
µ1
¯1 = 1 X n1 ¯2 = 1 X n2
µ2
n1
X 1 j
S c1 =
j=1
1 n1
n2
X 2 j
S c2 =
j=1
¯1 X
∈ N d (µ1, Σ/n1) ¯ 2 ∈ N d (µ2 , Σ/n2 ) X
− 1 j=1 1
n2
n1
− 1 j=1
− 1) S c1 ∈ (n2 − 1) S c2 ∈
− X ¯2 ∈ N d µ1 − µ2, (n1 − 1) S c1 + (n2 − 1) S c2 ∈
S c1
(n1
X 2 j
− X ¯2
d
(Σ, n1
1 1 + n1 n2
X 1 j
− X ¯1
X 2 j
− X ¯2
− 1) d (Σ, n2 − 1)
S c =
− X ¯1
n2
(n1
¯1 X
X 1 j
d
Σ ,
(Σ, n1 + n2
− 2)
− 1) S c1 + (n2 − 1) S c2 n1 + n2 − 2
S c2
Σ H 0 : µ1 = µ2 n1 n2 ¯1 X n1 + n2
− X ¯2 S c−1
X 11 , . . . , X1n
1
... X k1 , . . . , Xkn k
∈ ∈
¯1 X
− X ¯2 ∈ Γ2 (d, n1 + n2 − 2)
N d (µ1 , Σ1 ) ... N d (µk , Σk )
k H 0 : Σ1 =
· ·· = Σk k
sup
log
µ1 ,Σ1 ,...,µk ,Σk k
i=1
k
L (µi , Σi ) =
i=1
−
=
k
ni d log(2π) 2
k
sup log L (µi , Σi ) =
i=1 µi ,Σi
| |−
Σ = Σ1 = k
sup
k
− −
log
µ1 ,...,µk ,Σ
= sup = sup Σ
Q=
ni d log(2π) 2
nd log(2π) 2
k i=1 ni S i
−2log
ni d = 2
−
supµ supµ
1
− n2
sup log L (µi , Σ) = sup
Σ i=1
ni log Σ + 2
k
| | − nd2
ni log S i
i=1
¯i , Σ log L X
Σ−1 S i
Σ−1
log Σ +
Q n
=
nd log(2π) 2
n Q log 2 n
Q/n
Σ
1 ,...,µk ,Σ
k i=1 L (µi , Σ) k i=1 L (µi , Σi )
,Σ1 ,...,µk ,Σk
x ¯2 =
1 2
k
136.00 102.58 51.96
113.38 88.29 40.71
=
ni log
S 2 =
k
− | ∼ k
S 1 =
Q = n log n
i=1
x ¯1 =
nd log(2π) 2
·· · = Σk
Σ i=1 µi
i=1
Σ i=1
− | | | | − − −
L (µi , Σ) = sup
k
i=1
1 ni log S i 2
−
¯ i , S i log L X
−
ni log S i
i=1
Q −1 S n i
|
χ2 d(d+1)(k−1) 1 2
432.58 259.87 161.67 164.57 98.99 63.87 132.99 75.85 35.82 47.96 20.75 10.79
nd 2
