多元统计分析第03章.pdf

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1、1n?!?oN?b?u?1n?!?oN?b?u?August 17,20061n?!?oN?b?u?3.1 A?O?!?C?g.?3.1 A?O?!?CCC?ggg.?1.?n?X?ggg.?Xi N1(i,2)(i=1,n),?p?,PX=X1.Xn,KX Nn(,2In),?=(1,n)0.1n?!?oN?b?u?3.1 A?O?!?C?g.?X?g.?k?e?(?:(?1?i=0(i=1,n),2=1?,K=X0X=nXi=1X2i 2(n)?i=0(i=1,n),26=1?,K12X0X 2(n)(?2?i6=0(i=1,n),X0X?%?2?.1n?!?oN?b?u?3.1 A?O?

2、!?C?g.?X?g.?k?e?(?:(?1?i=0(i=1,n),2=1?,K=X0X=nXi=1X2i 2(n)?i=0(i=1,n),26=1?,K12X0X 2(n)(?2?i6=0(i=1,n),X0X?%?2?.1n?!?oN?b?u?3.1 A?O?!?C?g.?3.1.1?n?X Nn(,In)(6=0)K?C?=X0X?lgd?n,?%=0=Pni=12i?2?,P?X0X 2(n,)X0X 2n()(?3?X Nn(0n,2In),A?,?rank(A)=r,K?g.X0AX/2 2(r)A2=A(A?).(?4?X Nn(,2In),A=A0,K12X0AX 2(r,),

3、?=0A/2 A2=A?rank(A)=r(r n)1n?!?oN?b?u?3.1 A?O?!?C?g.?3.1.1?n?X Nn(,In)(6=0)K?C?=X0X?lgd?n,?%=0=Pni=12i?2?,P?X0X 2(n,)X0X 2n()(?3?X Nn(0n,2In),A?,?rank(A)=r,K?g.X0AX/2 2(r)A2=A(A?).(?4?X Nn(,2In),A=A0,K12X0AX 2(r,),?=0A/2 A2=A?rank(A)=r(r n)1n?!?oN?b?u?3.1 A?O?!?C?g.?3.1.1?n?X Nn(,In)(6=0)K?C?=X0X?lg

4、d?n,?%=0=Pni=12i?2?,P?X0X 2(n,)X0X 2n()(?3?X Nn(0n,2In),A?,?rank(A)=r,K?g.X0AX/2 2(r)A2=A(A?).(?4?X Nn(,2In),A=A0,K12X0AX 2(r,),?=0A/2 A2=A?rank(A)=r(r n)1n?!?oN?b?u?3.1 A?O?!?C?g.?(?5?g.?5?5:?X Nn(,2In),A?n?,B?m n?,-=X0AX,Z=BX,eBA=0,KBX?X0AX?p?.(?6?g.?p?:?X Nn(,2In),A,B?n?,K:AB=0 X0AX?X0BX?p?.1n?!?

5、oN?b?u?3.1 A?O?!?C?g.?(?5?g.?5?5:?X Nn(,2In),A?n?,B?m n?,-=X0AX,Z=BX,eBA=0,KBX?X0AX?p?.(?6?g.?p?:?X Nn(,2In),A,B?n?,K:AB=0 X0AX?X0BX?p?.1n?!?oN?b?u?3.1 A?O?!?C?g.?2.?p?ggg.p?g.?ke(?:(?1?X Np(,),0,KX01X 2(p,),?=01.(?2?X Np(,),0,A?,?rank(A)=r,K(X )0A(X )2(r)AA=A(?3?X Np(,),0,A,B?p?,K(X )0A(X )?(X )0B(

6、X )?AB=0pp1n?!?oN?b?u?3.1 A?O?!?C?g.?2.?p?ggg.p?g.?ke(?:(?1?X Np(,),0,KX01X 2(p,),?=01.(?2?X Np(,),0,A?,?rank(A)=r,K(X )0A(X )2(r)AA=A(?3?X Np(,),0,A,B?p?,K(X )0A(X )?(X )0B(X )?AB=0pp1n?!?oN?b?u?3.1 A?O?!?C?g.?2.?p?ggg.p?g.?ke(?:(?1?X Np(,),0,KX01X 2(p,),?=01.(?2?X Np(,),0,A?,?rank(A)=r,K(X )0A(X )

7、2(r)AA=A(?3?X Np(,),0,A,B?p?,K(X )0A(X )?(X )0B(X )?AB=0pp1n?!?oN?b?u?3.1 A?O?!?C?g.?3.?%t?%F?3.1.2?X N(,1)?Y 2(n)?p?,-T=XnY,KT?kn?gd?,?%?%t?,P?T t(n,).3.1.3?X 2(m,)?Y 2(n)?p?,-F=X/mY/n,KF?kgd?m,n,?%?F?,P?F F(m,n,).1n?!?oN?b?u?3.1 A?O?!?C?g.?4.?%2?,?%t?%F?AAAuO?1?a?V.1n?!?oN?b?u?3.1 A?O?!%?A(Wishart

8、)?!%?AAA(Wishart)?1.%?AAA?3.1.4?X()Np(0,)(=1,n)?p?,PX=(X(1),X(n)0?n p?,K?W=nX=1X()X0()=X0X?%?A?,P?W Wp(n,)?,?X()Np(,)(=1,n)?p?,PM=1n0W=X0X?l?%?%?A?,P?W Wp(n,),?=M0M=n01n?!?oN?b?u?3.1 A?O?!%?A(Wishart)?!%?AAA(Wishart)?1.%?AAA?3.1.4?X()Np(0,)(=1,n)?p?,PX=(X(1),X(n)0?n p?,K?W=nX=1X()X0()=X0X?%?A?,P?W W

9、p(n,)?,?X()Np(,)(=1,n)?p?,PM=1n0W=X0X?l?%?%?A?,P?W Wp(n,),?=M0M=n01n?!?oN?b?u?3.1 A?O?!%?A(Wishart)?1.%?AAA?555?555?1?X()Np(,)(=1,n)?p?,K?l?A?l%?A?,=A=nX=1(X()X)(X()X)0 Wp(n 1,)555?2ugd?n?k?5:?Wi Wp(ni,)(i=1,k)?p?,KkXi=1Wi Wp(n,)(n=n1+nk)555?3?p?W Wp(n,),Cm p?,Km?CWC0?l%?A?,=CWC0 Wm(n,CC0)1n?!?oN?b?

10、u?3.1 A?O?!%?A(Wishart)?1.%?AAA?555?555?1?X()Np(,)(=1,n)?p?,K?l?A?l%?A?,=A=nX=1(X()X)(X()X)0 Wp(n 1,)555?2ugd?n?k?5:?Wi Wp(ni,)(i=1,k)?p?,KkXi=1Wi Wp(n,)(n=n1+nk)555?3?p?W Wp(n,),Cm p?,Km?CWC0?l%?A?,=CWC0 Wm(n,CC0)1n?!?oN?b?u?3.1 A?O?!%?A(Wishart)?1.%?AAA?555?555?1?X()Np(,)(=1,n)?p?,K?l?A?l%?A?,=A=n

11、X=1(X()X)(X()X)0 Wp(n 1,)555?2ugd?n?k?5:?Wi Wp(ni,)(i=1,k)?p?,KkXi=1Wi Wp(n,)(n=n1+nk)555?3?p?W Wp(n,),Cm p?,Km?CWC0?l%?A?,=CWC0 Wm(n,CC0)1n?!?oN?b?u?3.1 A?O?!%?A(Wishart)?555?4%?A?:?X()Np(0,)(=1,n)?p?,?=11122122#rp rqfi?W=W11W12W21W22#rp r Wp(n,)K(1).W11 Wr(n,11),W22 Wpr(n,22)(2).?12=0?,W11?W22?p?.

12、1n?!?oN?b?u?3.1 A?O?!%?A(Wishart)?555?5?W Wp(n,),PW221=W22 W21W111W12,KW221 Wpr(n r,221),?221=22 2111112,?W221?W11?p?.555?6?W Wp(n,),KE(W)=n555?7(?)555?8(?)1n?!?oN?b?u?3.1 A?O?!%?A(Wishart)?555?5?W Wp(n,),PW221=W22 W21W111W12,KW221 Wpr(n r,221),?221=22 2111112,?W221?W11?p?.555?6?W Wp(n,),KE(W)=n555?

13、7(?)555?8(?)1n?!?oN?b?u?3.1 A?O?n!A?(Hotelling)T2?nnn!AAA?(Hotelling)T2?1.AAA?T2?3.1.5?X Np(0,),?W Wp(n,)(0,n p),?X?W?p?,K?O?T2=nX0W1X?AAA?T2?OOO?,?llln?gggddd?T2?,P?T2 T2(p,n)?,eX Np(,),KT2?%AAA?T2?,P?T2 T2(p,n,)1n?!?oN?b?u?3.1 A?O?n!A?(Hotelling)T2?1.AAA?T2?555?555?1?oNNp(,),?X()(=1,n),X?A O?l?,K?

14、O?T2=n(n 1)(X )0A1(X )T2(p,n 1)555?2T2?F?X:?T2 T2(p,n),Kn p+1npT2 F(p,n p+1)1n?!?oN?b?u?3.1 A?O?n!A?(Hotelling)T2?1.AAA?T2?555?555?1?oNNp(,),?X()(=1,n),X?A O?l?,K?O?T2=n(n 1)(X )0A1(X )T2(p,n 1)555?2T2?F?X:?T2 T2(p,n),Kn p+1npT2 F(p,n p+1)1n?!?oN?b?u?3.1 A?O?n!A?(Hotelling)T2?555?3?oNNp(,),?X()(=1,n

15、),X?A O?l?.PT2=n(n 1)X0A1XKn ppT2n 1 F(p,n p,)?=n01555?4T2?O?p,nk,?.555?5T2?O?zC?C.1n?!?oN?b?u?3.1 A?O?n!A?(Hotelling)T2?555?3?oNNp(,),?X()(=1,n),X?A O?l?.PT2=n(n 1)X0A1XKn ppT2n 1 F(p,n p,)?=n01555?4T2?O?p,nk,?.555?5T2?O?zC?C.1n?!?oN?b?u?3.1 A?O?n!A?(Hotelling)T2?555?3?oNNp(,),?X()(=1,n),X?A O?l?.P

16、T2=n(n 1)X0A1XKn ppT2n 1 F(p,n p,)?=n01555?4T2?O?p,nk,?.555?5T2?O?zC?C.1n?!?oN?b?u?3.1 A?O?o!%?d(Wilks)?O?9?ooo!%?ddd(Wilks)?OOO?999?1.%?ddd?3.1.6?X Np(,),K?1?“|?X?222?.e?X()(=1,n),A?l?,K|1nA|1n1A|?222?.3.1.7?A1 Wp(n1,),A2 Wp(n2,),0,n1 p,?A1?A2?,K2?=|A1|A1+A2|?%?ddd?OOO?OOO?,?%?ddd?.P?(p,n1,n2).1n?!

17、?oN?b?u?3.1 A?O?o!%?d(Wilks)?O?9?2.?OOO?T2F?OOO?XXX(?1?n2=1?,?n1=n p,K(p,n,1)d=11+1nT2(p,n)(?2?n2=2?,?n1=n pKn p+1p1 p(p,n,2)p(p,n,2)d=F(2p,2(n p+1)(?3?p=1?,Kn1n21 (1,n1,n2)(1,n1,n2d=F(n2,n1)(?4?p=2?,Kn1 1n21 p(2,n1,n2)p(2,n1,n2)d=F(2n2,2(n1 1)(?5?n2 2,p 2?,?2?O?F?O?Cq.1n?!?oN?b?u?3.1 A?O?o!%?d(Wilk

18、s)?O?9?3.?(?(?1e (p,n1,n2)K?3Bk(n1p+k2,n22)(k=1,p)?p?,?d=B1B2Bp.(?2en2=2(p)1n?!?oN?b?u?3.2?oN?u?9?&?!?u?2.?uuu?,T2=n(n 1)(X 0)0A1(X 0)T2(p,n 1)u?O?F=(n 1)p+1(n 1)pH0eF(p,n p)?wY,?F F(p,n p)1n?!?oN?b?u?3.2?oN?u?9?&?!q,?O?!qqq,?OOO?p?oNX?f(x,),.?b?flKH0:0,H1:/0r?X()(=1,n)?L(x(1),x(n);)=nYt=1f(x(t);)P?

19、L(X;)?qqq,.?O?=max0L(X;)/maxL(X;)?qqq,?OOO?.1n?!?oN?b?u?3.2?oN?u?9?&?!q,?O?d?q,?n?:XJ?,H0?*?d?X(t)(t=1,n)?VH0?*?d?X(t)(t=1,n)?V?.?knd?H0?,lq,?u,?u?flK?W=.nnn3.2.1?N?n?,2ln=2lnmax0L(X;)/maxL(X;)Cq?lgd?f?2?,f=?0?.1n?!?oN?b?u?3.2?oN?u?9?&?!q,?O?ub?flKH0:=0 H1:6=0?,?q,?O?=(|A|A0|)n/2?A0=nXi=1(X(i)0)(X(i

20、)0)0A=nXi=1(X(i)X)(X(i)X)01n?!?oN?b?u?3.2?oN?u?9?&?!q,?O?y?|A|A0|=11+1n1T2?T2=n(n 1)(X 0)0A1(X 0)H0eT2(p,n 1)?T2 F F?F=n ppT2n 1H0eF(p,n p)q,u?c(J?!1n?!?oN?b?u?3.2?oN?u?9?&?n!?&?&mnnn!?&?&mmm1.?&?dc?T2=n(X )0S1(X )T2(p,n 1)F=n p(n 1)pT2 F(p,n p)K?&?1?&?T2=n(X )0S1(X )(n 1)pn pFT?&?%3X?.1n?!?oN?b?u?3

21、.2?oN?u?9?&?n!?&?&m2.?&mmm?X Np(,),a Rp,kZ=a0X N(a0,a0a)?a?2z=a0a?,?t?O?t=n(a0X a0)a0Sa?a0?&?1?&m?a0X t/2a0San a0 a0X+t/2a0San?a=ei=(0,1,0)0K?i?&?1?&m xi t/2rsiin i xi+t/2rsiin1n?!?oN?b?u?3.2?oN?u?9?&?n!?&?&ma0?&mmmnnn3.2.2b?X(t)(t=1,n)?5gp?oNNp(,)(0,?)?,Kk?a,ma0X d,a0X+d(?d=s(n 1)pn(n p)Fa0Sa)?a0?V

22、?1 .T2mmmi?&?1?T2m?xi crsiin i xi+crsiin?c=s(n 1)p(n p)F?sii?S?1i?.1n?!?oN?b?u?3.3 oN?u?!?oN?u?!?oooNNN?uuu?X()(=1,n)?5goNX Np(1),)?Y()(=1,m)?5goNY Np(2),)?,?p?.?b?flKH0:(1)=(2),H1:(1)6=(2).1n?!?oN?b?u?3.3 oN?u?!?oN?u?1.?oooNNN?(?)?uuu?(1=2)u?O?T2=nmn+m(X Y)0(A1+A2n+m 2)1(X Y)?A1?A2?oN?l?.3H0?,T2 T2

23、(p,n+m 2).F=(n+m 2)p+1(n+m 2)pT2 F(p,n+m p 1)?W=F F(p,n+m p 1)1n?!?oN?b?u?3.3 oN?u?!?oN?u?2.?oooNNN?uuu?(16=2)(1)?m=n?,-Z(i)=X(i)Y(i)(i=1,n)?oN=z?oN?u?H0:(1)=(2)H0:Z=0p1n?!?oN?b?u?3.3 oN?u?!?oN?u?(2)?m 6=n?,?n m:-Z(i)=X(i)rnmY(i)+1nmnXj=1Y(j)1mmXj=1Y(j)(i=1,2,n)?y:E(Z(i)=(1)(2)Cov(Z(i),Z(j)=(1+nm2,?

24、i=j;0?i 6=j;def=ZijZ(i)Np(1)(2),Z)(i=1,n),?p?.=z?oN?u?!1n?!?oN?b?u?3.3 oN?u?!?oN?u?!?oooNNN?uuu?kk?oNNp(t),)(t=1,k),?X(t)()(t=1,k,=1,nt)?p?,?b?flKH0:(1)=(2)=(k),H1:?3i 6=j,?(i)6=(j)Pn=kXt=1nt,X=1nkXt=1ntXj=1X(t)(j),X(t)=1ntntXj=1X(t)(j)(t=1,k)1n?!?oN?b?u?3.3 oN?u?!?oN?u?ol?T=kXi=1niXj=1(X(i)(j)X)(X(i)(j)X)0|Sl?A=kXi=1niXj=1(X(i)(j)X(i)(X(i)(j)X(i)0=kXi=1Ai|ml?B=kXi=1ni(X(i)X)(X(i)X)0l?)“T=A+B1n?!?oN?b?u?3.3 oN?u?!?oN?u?q,?n?H0?u?O?=|A|A+B|=|A|T|?1Ai Wp(ni 1,)?p?,(i=1,k),d?5?A Wp(n,)23H0e,T Wp(n 1,).3?y3H0e,B Wp(k 1,),?B?A?p?.3H0e,u?O?(p,n k,k 1)?W=

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