(3.2)--线性模型习题课（习题二a).pdf

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1、5?.?SK1SK3.13?X1,X2,.,XnplN(0,2),yX=1nPni=1XiQ=Pn1i=1(XiXi+1)yX=1nPni=1Xi=1n10X-A=1n10,KX=AXQ=Pn1i=1(XiXi+1)=X1 Xn=1,0,.,0,1X,-B=1,0,.,0,1,kQ=BX,Cov(X,Q)=Cov(AX,BX)=ACov(X)B0,Xi N(0,2)pX Nn(0,2I)=Cov(X)=2I,ACov(X)B0=2AB0=2n?111.1?10.01=0,=Cov(X,Q)=0,XQ3.15?X Np(,I),Q1=X0AX,Q2=X0BX,eQ1Q2A 0,B 0,KAB=0

2、.yA 0,B 0,3?A1,B1,?A=A01A1,B=B01B1,Q1=X0AX=X0A01A1X=kA1Xk2,Q2=X0BX=X0B01B1X=kB1Xk2,Q1Q2A1XB1X?Cov(A1X,B1X)=A1Cov(X)B01=0,?Cov(X)=I,A1B01=0 A01A1B01B1=0 AB=0.4.35?.y=X+e,e (0,2),0.A?A?BLU?OpAn p?ByA?OyCov(By)Cov(A).pM1 M2M1 M2 0.y:A?A?BLU?O,k=(X01X)X01y,Cov(A)=A(X01X)X01Cov(y)1X(X01X)A0=A(X01X)X0121X

3、(X01X)A0=12A(X01X)X01X(X01X)A0=2A(X01X)A0,ByA?OE(By)=A?BX=A,?KBX=A.Cov(By)Cov(A)=2BB0 2A(X01X)A0=2BB0 2BX(X01X)X0B0=2B(X(X01X)X0)B0=2B12(I 12X(X01212X)X012)12B0,-Q=12X,=2B12(I Q(Q0Q)Q0)12B0=2B12(I PQ)(B12)0=2B12(I PQ)(B12(I PQ)0 0,Cov(By)Cov(A)0.4.45?.y=X+e,e (0,2I),rk(Xnp)=p,=(X0X)1X0y,2=ky Xk2/(n

4、p).(1)V ar(2).(2)?A=(I XX+)/(n p+2),OE(y0Ay 2)2.(3)yy0Ay2?O 2k?=MSE(y0Ay)MSE(2).)(1)2=kyXk2np=y0(IPX)ynp=e0(IPX)enp,V ar(2)=1(np)2V ar(e0(I PX)e)=1(np)2(m4 34)a0a+24tr(I PX)=1(np)2(m4 34)a0a+24(n p)=(m434)a0a(np)2+24np,aI PX?|?,m4=Ee4i.(2)E(y0Ay 2)2=V ar(y0Ay 2)+E(y0Ay 2)2,A=IXX+np+2y0Ay=1np+2(X+e)0

5、(IXX+)(X+e)=1np+20X0(IXX+)e+e0(IXX+)e=1np+20(IXX+)X0+e0Ae=e0Ae,y0Ay=e0Ae,E(y0Ay)=E(e0Ae)=1np+2E(e0(I XX+)e)=1np+22(np),E(y0Ay2)2=1np+22(np)22=44(np+2)2,V ar(y0Ay)=V ar(e0Ae)=1(np+2)2V are0(I XX+)e=1(np+2)2(m434)b0b+24(np),bI XX+?|?,m4=Ee4i.E(y0Ay 2)2=1(np+2)2(m4 34)b0b+24(n p)+44(np+2)2=(m434)b0b(np

6、+2)2+24np+2.(3)MSE(y0Ay)=E(y0Ay2)2=(m434)b0b(np+2)2+24np+2,MSE(2)=E(22)2=V ar(2)=(m434)a0a(np)2+24np,IPx=IX(X0X)X0=IX(X0X)+X0=IXX+,a=b.f?MSE(y0Ay)0,|?(C0iCi)2C0iACi C0iA1Ci 1 x0ixi C0iA1Ci1x0ixi C0iA1Ci,V ar(i)=2C0iA1Ci2/x0ixi.A=X0X=x01.x0p?x1.xp?=x01x1x01x2.x01xpx02x1x02x2.x02xp.x0px1x0px2.x0pxp|A|=Ppj=1x0ixjAij Aii=|A|Ppj6=ix0ixjAijx0ixi,KC0iA1Ci=Aii|A|=1Ppj6=ix0ixjAij/|A|x0ixi75?C0iA1Ci=1x0ixi1 Ppj6=ix0ixjAij/|A|=1,?Ppj6=ix0ixjAij/|A|=0,Aij6=0,x0ixj=0,i 6=j.5?x0ixj=0C0iA1Ci=1x0ixiV ar(i)=2C0iA1Ci=2/x0ixi.