Dayu Wu Applied Statistics Lecture Notes Estimation:MLE 1.Y~N(XB,021n) 2.L=(2m)(a2)-号ep{-克(Y-X8)T∑-1(Y-X8)} Propositions 1.8=(☒X)-lxTY 2.E(8=8 3.Var(8)=2(Xrz)-1 4.Gauss-Markov:E(Y)=XB,Var(Y)=a2I 5.Cou(8,e)=0 6.Y~N(XB,21) 7.a~N(8,g2(xx)-1) 8.要~X-p-1 Test 1.:==A=0,F=sg~Fp,n-p-) 2.o:8=0,6=高心n-p-1 3.F=号 4.(1-a)CIof月：(8-ta2vVGo,月+ta2vca) Standardization 1坊瑞 2斯=瑞 3两=2 Correlation 1.r 2 5 of 6 Dayu Wu Applied Statistics Lecture Notes Estimation: MLE 1. Y ∼ N(Xβ, σ2 In) 2. L = (2π) − n 2 (σ 2 ) − n 2 exp{− 1 2σ2 (Y − Xβ) TΣ −1 (Y − Xβ)} Propositions 1. βb = (X TX) −1X TY 2. E(βb) = β 3. V ar(β) = σ 2 (X TX) −1 4. Gauss-Markov: E(Y) = Xβ, V ar(Y) = σ 2 In 5. Cov(β, e b ) = 0 6. Y ∼ N(Xβ, σ2 In) 7. βb ∼ N(β, σ2 (X TX) −1 ) 8. SSE σ2 ∼ χ 2 n−p−1 Test 1. H0 : β1 = · · · = βn = 0, F = SSR/p SSE/(n−p−1) ∼ F(p, n − p − 1) 2. H0 : βj = 0, tj = βcj √cjjσb ∼ tn−p−1 3. Fj = t 2 j 4. (1 − α) CI of βj : (βbj − tα/2 √cjjσ, b βbj + tα/2 √cjjσb) Standardization 1. x ∗ ij = x√ ij−x¯j Ljj 2. y ∗ i = √ yi−y¯ Lyy 3. β ∗ j = √ Ljj √ Lyy βbj Correlation 1. r =   1 r12 . . . r1n r21 1 . . . r2n . . . . . . . . . . . . rn1 rn2 . . . 1   2. r 2 y1;2 = SSE(x2)−SSE(x1,x2) SSE(x2) 5 of 6