Dayu Wu Applied Statistics Lecture Notes Estimation: 1.可=a+月1 2.Unbiased:E=8o+B=Ey 3.Var(俞=Var(a)+x2var(a)+2rCou(a,属)=（任+=2）2 4.前~N(+x,(日+) T test 1.o:a=0,房N(0,) 2.=2∑c=2∑(-) 3t=高= F test 1.SST=∑(h-列2=Lw 2.SSR=∑(-2-会~X好 3.SSE=∑(-)2=(n-2)~X-2 4.SST SSR+SSE 5.F=胎心a-2 Correlation 1r=aV儡 2.2-器=是 3.R2=1- 4.t=心tn- Residual 1.Ee:=0 2.Var(e)=Var(-)=（1-A-）o2 3.Leverage:h 3of6 Dayu Wu Applied Statistics Lecture Notes Estimation: yb 1. yb = βb0 + βb1x 2. Unbiased: Eyb = β0 + β1x = Ey 3. V ar(yb) = V ar(βb0) + x 2V ar(βb1) + 2xCov(βb1, βb0) =  1 n + (x−x¯) 2 Lxx  σ 2 4. yb ∼ N(β0 + β1x,  1 n + (x−x¯) 2 Lxx  σ 2 ) T test 1. H0 : β1 = 0, βb1 ∼ N(0, σ 2 Lxx ) 2. σb 2 = 1 n−2 Pe 2 i = 1 n−2 P(yi − ybi) 2 3. t = βb √ 1 σb2/Lxx = βb1 √ Lxx σb F test 1. SST = P(yi − y¯) 2 = Lyy 2. SSR = P(ybi − y¯) 2 = L2 xy Lxx ∼ χ 2 1 3. SSE = P(yi − ybi) 2 = (n − 2)σb 2 ∼ χ 2 n−2 4. SST = SSR + SSE 5. F = SSR/1 SSE/(n−2) ∼ F1,n−2 Correlation 1. r = √ Lxy LxxLyy = βb1 qLxx Lyy 2. r 2 = SSR SST = L2 xy LxxLyy 3. R2 = 1 − SSE SST 4. t = √ n−2r √ 1−r 2 ∼ tn−2 Residual 1. Eei = 0 2. V ar(ei) = V ar(yi − ybi) =  1 − 1 n − (x−x¯) 2 Lxx  σ 2 3. Leverage: hi = 1 n + (x−x¯) 2 Lxx 3 of 6