17 5 13 1 2 1.1... 2 1.2... 2 1.3... 3 2 3 2.1... 3 2.2... 5 3 6 3.1... 6 3.2... 7 3.3 t... 7 3.4 BC a... 9 3.5... 10 4 11 1
1 θ n ˆθ. ˆθ, ˆθ, ˆθ.,, ˆθ.,.,,,. 1.1 ˆθ σ 2 = E(ˆθ E ˆθ) 2 b = E(ˆθ θ). Y 1,,Y n f(y θ), ˆθ, ˆσ 2 =1/(nI(ˆθ)) n σ 2., I(θ) =E( log f(y θ)/ θ) 2 1. ˆσ 2 σ 2 = 1/ n l(y i=1 i ˆθ)., l(y θ) = 2 log f(y θ)/ θ 2. ˆθ,., (U 1,V 1 ),, (U n,v 1 ) ˆθ = Ū/ V θ = EU/EV., Ū = n i=1 U i/n, V = n i=1 V i/n., ˆθ ˆσ 2 = 1 (Ū ) 2 { S 2 u n V + (Ū)2 S2 v ( V ) 2 2S uv Ū V. Su 2 = n 1 n j=1 (U j Ū) 2, Sv 2 = n 1 n j=1 (V j V ) 2, S uv = n 1 n j=1 (U j Ū)(V j V )., ˆθ.,., ˆθ (4nf 2 (ˆθ)) 1,, f(θ).,,., ˆθ = Ū/ V,, ( 1 Ū S 2 ˆb = u n V Ū S ) uv (2) 2 Ū V. } (1) 1.2 θ ˆθ = ˆθ(Y 1,,Y n )., Y 1,,Y n, ˆθ Y 1,,Y n 2
. i Y i, ˆθ (i) = ˆθ(Y 1,,Y i 1,Y i+1,,y n ), ˆθ ( ) = n 1 n i=1 ˆθ (i), ˆbJ =(n 1)(ˆθ ( ) ˆθ). (3), θ = ˆθ ˆb J = nˆθ (n 1)ˆθ ( ) (4) ˆθ.,, ˆθ. E ˆθ = θ + a 1 /n + a 2 /n 2 + (5) a 1,a 2, n, b = a 1 /n + a 2 /n 2 + ˆθ., n 1 ˆθ (i),. E ˆθ (i) = θ + a 1 /(n 1) + a 2 /(n 1) 2 + (6) (5) (6), θ. E θ ] = E [nˆθ (n 1)ˆθ ( ) = θ + a 2 /(n(n 1)) + (7), θ a 2 /(n(n 1))+ = O(1/n 2 ) ˆθ a 1 /n+a 2 /n 2 + = O(1/n),. 1.3 ˆθ = Ȳ., ˆθ (i) =(nˆθ Y i )/(n 1), ˆθ( ) = ˆθ, ˆθ(i) ˆθ ( ) =(Ȳ Y i)/(n 1). ˆσ J 2 = n 1 n (ˆθ (i) n ˆθ ( ) ) 2 (8) i=1, ˆθ = Ȳ, ˆσ J 2 = n 1 n i=1 (Y i Ȳ ) 2 /(n 1), ˆθ σ 2 = n 1 Var(Y 1 )., ˆθ, (8) ˆσ J 2 ˆθ., ˆθ Y 1,,Y n. 2 2.1 F (y) Y 1,,Y n, n F n (y) =n 1 δ(y i y) 3 i=1
., δ(y i y), Y i y 1, 0. 4.6 n =20., Glivenko-Cantelli, n, F n (y) F (y)., E F F = F (y), ˆθ = ˆθ(Y 1,,Y n ) σ 2 = E F (ˆθ E F ˆθ) 2., F, σ 2., n, F F n = F n (y), ˆσ 2 = E Fn [ˆθ(Y 1,,Y n ) E F n ˆθ(Y 1,,Y n ) ] 2 (9) σ 2. Y 1,,Y n F n,. ˆθ = ˆθ(Y 1,,Y n ). 9 ˆσ 2,. 11.1 ˆθ = Ȳ = n j=1 Y j/n. ˆθ σ 2 = n 1 var Y 1 var Y 1 S 2 = n j=1 (Y j Ȳ )2 /n σ 2 n 1 S 2 /n, Ȳ Ȳ =(Y1 + + Yn )/n, E Fn Ȳ = Ȳ., ˆσ 2 = E Fn (Ȳ Ȳ ) 2 = n 1 S 2 n,,.,,,. 11.1 ( ) (i) y 1,,y n n n Y1 b,,y b n, ˆθ b b = ˆθ(Y 1,,Y b n ) (ii) (i) B ˆθ 1,, ˆθ B (iii) ˆσ 2 b = B (ˆθ b ˆθ ) 2 /(B 1). (10) b=1, ˆθ = B 1 B b=1 ˆθ b. (9) ˆσ 2 ˆσ 2 b B n, n B. 4
11.1 (, 1981) 11.1 (1/8 ) 188 96 168 176 153 172 177 163 146 173 186 168 177 184 96 161.53 27.95 139 163 160 160 147 149 149 122 132 144 130 144 102 124 144 140.60 15.86 11.1, (1) 11.2 B,,., ˆσ J 2 > ˆσ2 b. B 11.2 ( 11.1), B 100 0.0044 200 0.0049 500 0.0051 0.0053 0.0049 1000 0.0048 2000 0.0050 100 0.0383 200 0.0385 500 0.0418 0.0503 0.0291 1000 0.0385 2000 0.0371 2.2, b = E F ˆθ θ ˆb = EFn ˆθ(Y 1,,Y n ) ˆθ (11) 5
., ˆb ˆbb = B 1 B b=1 ˆθ(Y b 1,,Y b n ) ˆθ. (12) 11.2 ˆθ = n 1 n j=1 (Y j Ȳ ) 2 ˆθ b = σ 2 /n F n ˆθ [ ] n ˆb = EFn n 1 (Yj Ȳ ) 2 ˆθ j=1 = (n 1)n 1ˆθ ˆθ = n 1ˆθ ˆθ ˆθ, θ = ˆθ ˆb.,.,,,,. 3 3. 2 t 3 BC a 3.1 n, θ ˆθ. ˆσ 2 /n ˆθ,, n T = n(ˆθ θ)/ˆσ N(0, 1) 1 { } Pr n(ˆθ θ)/ˆσ t =Φ(t)+O(n 1/2 ) (13) 6
Φ(t) N(0, 1), O(n 1/2 ) n 1/2. z α N(0, 1) 100α Φ(z α )=α (13) Pr{θ ˆθ n 1/2ˆσz α } =1 α + O(n 1/2 ) (14), I =(, ˆθ + n 1/2ˆσz 1 α ) (14), Pr {θ I L } (1 α) =O(n 1/2 ). (15) 1 α I, 1 t, BC a I L Î 2.,. Pr{θ Î} (1 α) =O(n 1 ) (16) 3.2 (14) 1 n θ 1 α. 11.2 ( ) (i) y 1,,y n Y1,,Y n (ii) ˆθ = ˆθ(Y 1 ˆθ 1,, ˆθ B (iii) ˆθ α = ˆθ (αb) αb ˆθ (αb) ˆθ 1,, ˆθ B αb (ˆθ α, ) θ 1 α,,y n ) B 3.3 t ˆθ 7
t ˆθ ˆσ/ n (13) T = n(ˆθ θ)/ˆσ 1 T T 2 t 2 Pr{T t} = Φ(t)+n 1/2 p(t)φ(t)+o(n 1 ), Pr{T t} = Φ(t)+n 1/2ˆp(t)φ(t)+O p (n 1 ). p(t) 2 ˆp(t) p(t) Φ(t), φ(t), Pr{T t} Pr{T t} = O p (n 1 ) (17) Pr{T t} 2 T 100α w α T 100α 2 z α 1 Pr {θ ˆθ n 1/2ˆσw } α =1 α + O(n 1 ) (18) t (, ˆθ n 1/2ˆσw α ) 2 t (14) z α (18) T w α t 11.3 ( t ) (i) y 1,,y n Y 1,,Y n (ii) (i) ˆθ ˆσ ˆθ ˆσ (iii) t T = n(ˆθ ˆθ)/ˆσ B T1,,TB (iv) w α = T(αB) αb, T(αB) 11.2. (, ˆθ n 1/2ˆσw α ) 1 α 8
3.4 BC a, ˆθ, ˆθ, 1 BC a 2 z 0 a ˆθ ˆθ ˆθ z 0 =Φ 1 (Pr[ˆθ ˆθ]) ˆθ θ 1 Φ ˆθ ˆθ Pr[ˆθ ˆθ] =0.5 z 0 =0 z 0 ˆθ 1,, ˆθ B ẑ 0 =Φ ( {ˆθ 1 b < ˆθ}/B ) (19) a. ˆθ ( (2003) )) a ˆθ (i), i y i ˆθ, ˆθ ( ) = n 1 n ˆθ i=1 (i). a { n } 3/2 â = (ˆθ ( ) ˆθ n (i) ) 2 (ˆθ ( ) ˆθ (i) ) 3 /6. (20) i=1 11.4 (BC a ) 1. y 1,,y n Y1,,Yn 2. ˆθ = ˆθ(Y 1,,Yn ) ( B ) ˆθ 1,, ˆθ B,. 3. ẑ 0 â, (19) (20) 4. (ˆθ (ˆαB), ) θ 1 α α (0 < α<1) ˆα ( ) ẑ 0 + z α ˆα =Φ ẑ 0 +. (21) 1 â(ẑ 0 + z α ) 1 2α (ˆθ (ˆαB), ˆθ (d 1 αb) ), i=1 1 α (21) z α z 1 α BC a BC a 2 BC a ˆθ (ˆαB), ˆθ (d 1 αb) (21) â =0 ˆα =Φ(z α +2ẑ 0 ) â ẑ 0 0 ˆα = α 9
3.5 11.3 n =8, X i =, Y i =, i =1,,n, r = E(Y )/E(X). r 1 α [ a, a] (a>0), (α, a) =(0.1, 0.2) 11.3 (Efron & Tibshirani (1993, p.373) ) 1 17649 16449 9243 2 12013 14614 9671 3 19979 17274 11792 4 21816 23798 13357 5 13850 12560 9055 6 9806 10157 6290 7 17208 16570 12412 8 29044 26325 18806 11.3, ˆr =ȳ/ x = 0.0713 [ 0.2, 0.2], 11.1 ( ) B = 2000 ˆr, 11.1 ( ) 90% ( 0.208, 0.105), [ 0.2, 0.2] 2., BC a. 11.1 ( ) ẑ 0 =Φ { 1 (ˆr b < ˆr)/B } =0.029, ẑ 0 > 0 ˆr (20) â =0.024 α =0.05 ẑ 0,â ˆα =0.0634, 1 α =0.9619 ˆα + 1 α 1 BC a ( 0.201, 0.131),., 11.1 ( ), B = 2000 t T = n(ˆr ˆr)/ˆσ. ˆσ 2 (1). 10
, 90% t ( 0.226, 0.319), BC a., 2 4 3 0.3 2 0.2 1 0.1-0.11 0.11 0.33-18.25-13.25-8.25-3.25 1.75 11.1 11.3 B = 2000 ˆr = Ȳ / X ( ) T = n(ˆr ˆr)/ˆσ (ˆr = 0.0713). 4 11.2 x, y µ x, µ y t =( x ȳ)/ s 2 x /(m 1) + s2 y /(n 1) µ x = µ y x = m 1 m i=1 x i,ȳ = n 1 n j=1 y j s 2 x = m 1 m i=1 (x i x) 2,s 2 y = n 1 n j=1 (y j ȳ) 2 t =2.44, µ x = µ y t.,. x i = x i x + z, i =1,,m, y j = y j ȳ + z, j =1,,n z =(m + n) 1 (m x + nȳ) m n F m (x) =m 1 δ(x i x), G n(y) =n 1 δ(y j y) i=1. F m (x) F n (y) z,.,, F m X1,...,Xm G n Y1,...,Yn, t =( X Ȳ )/ S 2 x /(m 1) + S 2 11 j=1 y /(n 1)
X, Ȳ,Sx 2,S 2 y, t, p, p =Pr{ t t }. α, p α,,,., p,. ˆp = 1 B B δ ( t b t ). (22) b=1 B = 10000, (22) 11.1, ˆp =0.0406 α =0.05 1996, 2003 [1] Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application, Cambridge University Press: Cambridge. [2] Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap, Chapman & Hall: New York. [3] (1990).,, 19, 137 162. [4] (1992).,, 21, 67 100. [5] (1993).,, 22, 257 312. [6], (1981). 2,,. [7],,, (1992).,, 19, 50 81. [8], (1996). 2,, 44, 3 18. [9], (2003)., 11, I,. 12