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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)
2、,X0X?%?2?.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?!?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) =
3、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?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,),? =
4、 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.?(?5?g.?5?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.?(?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.?
6、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 ) 2(r) AA = A(?3?X Np(,), 0,A,B?
7、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 ) 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?
8、,?%?%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)?!%?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(,)( =
9、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 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
10、?!%?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?
11、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 =nX=1(X()X)(X()X)0 Wp(n 1,)555?2ugd?n?k?5:?Wi Wp(
12、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?.1n?!?oN?b?u?3.1 A?O?!%?A(Wis
13、hart)?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?7(?)555?8(?)1n?!?oN?
14、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?O?T2=n(n 1)(X
15、)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,
16、,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?.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
17、, ,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?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?OO
18、O?,?%?ddd?. P? (p,n1,n2).1n?!?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
19、2?,?2?O?F?O?Cq.1n?!?oN?b?u?3.1 A?O?o!%?d(Wilks)?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
20、,),.?b?flKH0: 0,H1: / 0r?X()( = 1, ,n)?L(x(1), ,x(n);) =nYt=1f(x(t);)P?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?.1
21、n?!?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) 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.?&?
22、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?3.2 ?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/2rsi
23、in1n?!?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?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
24、),)?,?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(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
25、(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,?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?!?oooN
26、NN?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 =
