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1 t BC a

2 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

3 . 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 F (y) Y 1,,Y n, n F n (y) =n 1 δ(y i y) 3 i=1

4 ., δ(y i y), Y i y 1, 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, 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,,.,,, ( ) (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

5 11.1 (, 1981) 11.1 (1/8 ) , (1) 11.2 B,,., ˆσ J 2 > ˆσ2 b. B 11.2 ( 11.1), B , b = E F ˆθ θ ˆb = EFn ˆθ(Y 1,,Y n ) ˆθ (11) 5

6 ., 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.,.,,,, 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

7 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 ( ) (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

8 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) (, ˆθ n 1/2ˆσw α ) 1 α 8

9 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= (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

10 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) ) , ˆr =ȳ/ x = [ 0.2, 0.2], 11.1 ( ) B = 2000 ˆr, 11.1 ( ) 90% ( 0.208, 0.105), [ 0.2, 0.2] 2., BC a ( ) 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 = ˆα + 1 α 1 BC a ( 0.201, 0.131),., 11.1 ( ), B = 2000 t T = n(ˆr ˆr)/ˆσ. ˆσ 2 (1). 10

11 , 90% t ( 0.226, 0.319), BC a., B = 2000 ˆr = Ȳ / X ( ) T = n(ˆr ˆr)/ˆσ (ˆr = ) 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)

12 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 = α = , 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, [4] (1992).,, 21, [5] (1993).,, 22, [6], (1981). 2,,. [7],,, (1992).,, 19, [8], (1996). 2,, 44, [9], (2003)., 11, I,. 12

1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l

1 1 ϕ ϕ ϕ S F F = ϕ (1) S 1: F 1 1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l : l r δr θ πrδr δf (1) (5) δf = ϕ πrδr