2019 1 15 26 3.4, 13.4
BLUE t F 1
1. TSS ESS RSS TSS = ESS + RSS R 2 := ESS/TSS = 1 RSS/TSS R 2 := 1 [RSS/(n k)]/[tss/(n 1)] 2. y i ŷ i y i x i 2 2
3. 1 2 k F 1 F 4. 2 2 3
1 5 1.1 pp. 60, 272........... 5 1.2............ 13 1.3 pp. 63, 272.......... 16 2 20 2.1 1................ 20 2.2.............. 24 2.3 F............... 28 2.4 2................ 35 4
1 1.1 pp. 60, 272 y i x i E(y i x i ) = α + βx i ŷ i := a + b x i e i := y i ŷ i 5
1. e i = 0 x i e i = 0 6
OLS min a,b and (y i a bx i ) 2 a, b R 1 (y i a b x i ) = 0 x i (y i a b x i ) = 0 7
1. (y 1,..., y n ) TSS := (y i ȳ) 2 2. (y 1,..., y n ) ESS := (ŷ i ȳ) 2 8
3. (y 1,..., y n ) 2 RSS := e 2 i 2. TSS = ESS + RSS 9
TSS := = = = (y i ȳ) 2 [(ŷ i ȳ) + e i ] 2 [ (ŷi ȳ) 2 + 2(ŷ i ȳ)e i + e 2 i ] (ŷ i ȳ) 2 + 2 (ŷ i ȳ)e i + e 2 i 10
(ŷ i ȳ)e i = [(a + b x i ) (a + b x)]e i n = b (x i x)e i n = b x i e i b x = 0 e i 11
4. R 2 := ESS TSS R 2 = 1 RSS TSS 12
1.2 5. R 2 := 1 RSS/(n k) TSS/(n 1) 13
E ( 1 n 1 ) (y i ȳ) 2 = var(y i ) E ( 1 n k e 2 i ) = var(u i ) R 2 1 var(u i )/ var(y i ) 14
E ( ) ( n [1/(n k)] R2 = 1 E e2 i [1/(n 1)] n (y i ȳ) 2 1 E ( [1/(n k)] n ) e2 i E([1/(n 1)] n (y i ȳ) 2 ) = 1 var(u i) var(y i ) ) 15
1.3 pp. 63, 272 6. y i ŷ i y i x i y i (x i,1,..., x i,k ) 3. 2 16
(ŷ 1,..., ŷ n ) 1 n ŷ i = 1 n (a + b x i ) = a + b x = ȳ 17
((y 1, ŷ 1 ),..., (y n, ŷ n )) 1 n (y i ȳ)(ŷ i ȳ) = 1 n = 1 n = 1 n [(ŷ i ȳ) + e i ](ŷ i ȳ) (ŷ i ȳ) 2 + 1 n (ŷ i ȳ) 2 e i (ŷ i ȳ) 18
((y 1, ŷ 1 ),..., (y n, ŷ n )) (1/n) n (y i ȳ)(ŷ i ȳ) n (1/n) (y i ȳ) 2 (1/n) n (ŷ i ȳ) 2 (1/n) n = (ŷ i ȳ) 2 n (1/n) (y i ȳ) 2 (1/n) n = n (ŷ i ȳ) 2 n (y i ȳ) 2 (ŷ i ȳ) 2 = ESS TSS 19
2 2.1 1 2 k k N ( µ 1, σ 2),..., N ( µ k, σ 2) n 1,..., n k (y 1,1,..., y 1,n1 ),..., (y k,1,..., y k,nk ) n := n 1 + + n k 7. µ 1,..., µ k µ := µ 1 + + µ k k 20
8. µ 1,..., µ k 1. 9. h = 1,..., k 2. 10. α h := µ h µ h 3. 21
11. 1 h = 1,..., k, i = 1,..., n h y h,i = µ h + u h,i u h,i N ( 0, σ 2) α 1 + + α k = 0 y h,i = µ + α h + u h,i u h,i N ( 0, σ 2) 22
h = 1,..., k, i = 1,..., n h x h,i := h y h,i = k j=1 µ j [x h,i = j] + u h,i 23
2.2 h = 1,..., k ȳ h := 1 n h n h y h,i ȳ := 1 n k h=1 n h y h,i 24
12. S := 13. S w := k h=1 k h=1 n h n h (y h,i ȳ) 2 (y h,i ȳ h ) 2 25
14. S b := k h=1 n h (ȳ h ȳ) 2 4. S = S w + S b 26
S = = k h=1 k h=1 n h n h = S w + 2 2 0 (y h,i ȳ h + ȳ h ȳ) 2 [ (yh,i ȳ h ) 2 + 2(y h,i ȳ h )(ȳ h ȳ) + (ȳ h ȳ) 2] k (ȳ h ȳ) h=1 n h (y h,i ȳ h ) + S b 27
2.3 F H 0 : µ 1 = = µ k vs H 1 : µ h µ for some h = 1,..., k 5. S w σ 2 χ2 (n k) 28
h = 1,..., k nh (y h,i ȳ h ) 2 σ 2 χ 2 (n h 1) k h=1 nh (y h,i ȳ h ) 2 σ 2 χ 2 ( k ) (n h 1) h=1 29
6. H 0 S σ 2 χ2 (n 1) H 0 {y h,i } IN ( µ, σ 2) 7. S w S b 30
1. 1. X + Y χ 2 (m) 2. X χ 2 (n) 3. X Y Y χ 2 (m n) 31
X Y t M X+Y (t) = M X (t)m Y (t) M Y (t) = M X+Y (t) M X (t) = χ 2 (m n) mgf (1 2t) m/2 (1 2t) n/2 = (1 2t) (m n)/2 32
8. H 0 S b σ 2 χ2 (k 1) H 0 S σ 2 χ2 (n 1) S w σ 2 χ2 (n k) S w S b S σ 2 S w σ 2 χ2 (k 1) 33
1. H 0 S b /(k 1) S w /(n k) F(k 1, n k) 15. 1 F k 1 S b S b /(k 1) [S b /(k 1)]/[S w /(n n k S w S w /(n k) n 1 S S/(n 1) 34
2.4 2 2 A, B A j = 1,..., J B k = 1,..., K 16. 2 j = 1,..., J, k = 1,..., K, i = 1,..., n j,k y j,k,i = µ + α j + β k + γ j,k + u j,k,i u j,k,i N ( 0, σ 2) α j, β k, γ j,k 35
j = 1,..., J, k = 1,..., K, i = 1,..., n j,k x j,k,i := j z j,k,i = k y j,k,i = µ + + J j =1 J j =1 α j [x j,k,i = j ] + K k =1 K k =1 β j [z j,k,i = k ] γ j,k [x j,k,i = j ][z j,k,i = k ] + u j,k,i 36
1 1 2 37
3 4 13 4 (1) (2) (3) (4) 38