主成分分析における固有値・固有ベクトルの信頼区間 －漸近理論および ...

1
[1] P84 3
mvrnorm
[5]
[2][3][6]t [4]
2
M2007MM016
l
S
a S λ k α k
λ k = E(l k ) α k = E(a k ) z α
100α%
2.1
⎧
⎨ 2λ 2 k
k = k ′
cov(l k , l k ′) = n − 1
⎩
0 k ≠ k ′
l k
l k ∼ N(λ k ,
l k − λ k
N(0, 1)
λ k [2/(n − 1)]
1/2
2λ 2 k
n − 1 ) (1)
τ 2 = 2/(n − 1) λ k
(1 − 2α)
l k
l k
≤ λ k ≤
(2)
1 + τz α 1 − τz α
n
τz α ≤ 1
log l k
(1)
log l k ∼ N(log λ k ,
2
n − 1 ) (3)
λ k
1 − α
log λ k log l k ± τz α
λ k
l k e −τzα ≤ λ k ≤ l k e τzα (4)
l k λ k
(2) (4)
2.2
α kj
cov(a kh , a k ′ j ′) = ⎧⎪ ⎨
⎪ ⎩
λ k
n − 1
p∑
l=1,l≠k
λ l α lj α lj ′
(λ l − λ k ) 2 k = k ′
− λ kλ k ′α kj α k′ j ′
(n − 1)(λ k − λ k ′) 2 k ≠ k ′
a kj
λ k a kj
α kj
a k
a k N(α k , T k )
T k =
λ k
n − 1
p∑
l=1,l≠k
λ l
(λ l − λ k ) 2 α lα ′ l
T k a k 1
(p − 1)
(n − 1)α ′ k(l k S −1 + l −1
k S − 2I p)α k ≤ χ 2 (p−1);α (5)

a ki ∼ N(α ki , T k(i,i) ) (2)
√
√
a ki − T k(i,i) z α ≤ α ki ≤ a ki + T k(i,i) z α (6)
T k(i,i) a k T k i
3
t
BCa
3.1
[3]
P r{ √ n(ˆθ − θ)/ˆσ ≤ t} = Φ(t) + O(n −1/2 ) (7)
t BCa
P r{θ ∈ I L } − (1 − α) = O(n −1 ) (8)
(7) 1 (8)
2
3.2
α
B
Bα
3.2.1
(i). 1 y 1 , . . . , y n
n y ∗ 1, . . . , y ∗ n
(ii). ˆθ ∗ = θ(Y ∗
1 , . . . , Y ∗ n )
B ˆθ ∗ 1, . . . , ˆθ ∗ B
(iii). ˆθα = ˆθ (Bα) ∗ Bα ˆθ (Bα) ∗
ˆθ 1, ∗ . . . , ˆθ B ∗ Bα
(ˆθ α , ˆθ 1−α ) θ
1 − 2α
3.3 t
ˆθ
1
t
ˆθ
ˆσ/ √ n (7)
T = √ n(ˆθ − θ)/ˆσ
1
T ∗ T 2
(ˆθ − n −1/2ˆσw 1−α , ˆθ − n −1/2ˆσw α )
2 1 − 2α
3.3.1
t
[4]
˜θ n,i = nˆθ n − (n − 1)ˆθ (i)
n−1 ,
i = 1, . . . , n
ˆθ (i)
n−1
i ˆθ
˜θ n,i , i = 1, . . . , n
˜θ n,i √ nˆθ n
V ( √ nˆθ n )
1
n∑
(˜θ n,i − 1 n∑
˜θ n,j ) 2
n − 1 n
i=1
j=1
V (ˆθ n ) = V ( √ nˆθ n )/n
V JACK =
1
n(n − 1)
= n − 1
n
n∑
i=1
n∑
(˜θ n,i − 1 n
i=1
(ˆθ (i)
n−1 − 1 n
n∑
˜θ n,j ) 2 (9)
j=1
n∑
j=1
ˆθ (j)
n−1 )2 (10)

3.3.2
(i). 1 y 1 , . . . , y n
n y ∗ 1, . . . , y ∗ n
(ii). (i) ˆθ ˆσ
ˆθ ∗ ˆσ ∗
(iii). t T ∗ = √ n(ˆθ ∗ − ˆθ)/ˆσ ∗
B T ∗ 1 , . . . , T ∗ B
(iv). w α = T ∗ (Bα) Bα T ∗ (Bα)
(ˆθ − n −1/2ˆσw 1−α , ˆθ − n −1/2ˆσw α )
1 − 2α
3.4 BCa
ˆθ ˆθ
ˆθ ˆθ ∗
BCa
z 0 a
ˆθ ∗
ˆθ ˆθ ∗
ˆθ ∗ ˆθ z 0 = Φ −1 (P r[ˆθ ∗ ≤
ˆθ]) ˆθ θ
Φ
ˆθ ∗ ˆθ
P r[ˆθ ∗ ≤ ˆθ] 0.5 z 0 = 0 z 0
ˆθ ∗ = θ(Y ∗
1 , . . . , Y ∗ n )
3.4.1
(i). 1 y 1 , . . . , y n
n y ∗ 1, . . . , y ∗ n
(ii). ˆθ ∗ = θ(Y ∗
1 , . . . , Y ∗ n )
B ˆθ ∗ 1, . . . , ˆθ ∗ B
(iii). ẑ 0 â (11)
(12)
(iv). (ˆθ α , ˆθ 1−α ) = (ˆθ ∗ (B ̂α) , ˆθ ∗ (B̂1−α) ) θ
1−2α α(0 ≤
α ≤ 1) ˆα
ˆα = Φ(ẑ 0 +
ẑ 0 + z α
1 − â(ẑ 0 + z α ) ) (13)
̂1 − α (13) z α z 1−α
4
[1] P84 3
[1]
mvrnorm
50
200500 3
149 38.7 72.23333 79.36667
53.51724 40.79310 27.58621 28.75862
40.79310 41.73448 29.83103 24.35517
27.58621 29.83103 26.52989 17.22184
28.75862 24.35517 17.22184 18.24023
1
ẑ 0 = Φ −1 (#{ˆθ ∗ b ≤ ˆθ}/B) (11)
ẑ 0
(11) B B
a
ˆθ
ˆθ (i) i y i
ˆθ ˆθ () =
n −1 ∑ n
i=1 ˆθ (i) â
â =
∑ n
i=1 (ˆθ () − ˆθ (i) ) 3
6{ ∑ n
i=1 (ˆθ () − ˆθ (i) ) 2 } 3/2 (12)
124.814224 10.848728 2.495491 1.863396
-0.6240226 0.6455638 0.22363789 -0.37924835
-0.5591912 -0.3456392 -0.74576837 -0.10801960
-0.4083340 -0.6604700 0.62447014 -0.08414179
-0.3621661 0.1660133 0.06206998 0.91510798
2
5

5.1
2 1 (1,1)
1 124.814224
(1,1) −0.6240226
+1 2
1000
.945 .964 .969 .949
.943 .956 .973 .940
4
.946 .959 .919 .967
.952 .941 .820 .915
.954 .949 .857 .913
.960 .954 .908 .981
5
.957 .938 .956 .877
.941 .947 .948 .953
.944 .939 .905 .944
.934 .949 .917 .945
.935 .949 .948 .827
6
.952 .940 .912 .903
.940 .946 .957 .939
.943 .938 .885 .949
.936 .947 .938 .937
.934 .946 .930 .858
7 BCa
2 1 (1,1)
112.6412 157.1251
111.2681 154.7389
103.6404 158.4347
t 110.3003 174.1243
BCa 106.2877 160.8743
3 1
3 1 1
n 200
95% t BCa
1000
5.2
t
t
6
1000
n=200 1000
7
n
n
8
BCa t
[1] ,
(1983).
[2] 21 3:
, (2008).
[3]
11-,
(2003).
[4] ,
(2001).
[5] I.T. JolliffePrincipal Component Analysis (Springer
Series in Statistics) , Springer(2002).
[6] Efron.B,Tibshirani,R.J.An Introduction to the
Bootstrap. Chapman & Hall,New York(1993).