多元统计分析

11.1.,;,;.,.,,.,..,,.,.,.,.,,.,,..,.,,.,()()....,.,,.,.(,.)21.,...,.,.,,;,.tT2..,..,,.,,..,,,...,(),.().,..(),;..1.2,,.,.,.,Adrian(1808),Laplace(1811),Plana(1813),Gauss(1823)Bravais(1846).FrancisGal-ton,.[,Galton(1889).].1.23KarlPearson,.R.A.Fisher,.,,,.,,,,..,.,.,.,.;;,.,,.,,,.,.,,,,..,,.,,,,..,...,,.,..,,Walker(1931).41,,..,.,,.,..22.1.2.2().2.3,,.2.4,,.2.5,,,.().,.,.2.6.2.7,.2.22.2.1,.()XY.(cdf).(x;y),cdfF(x;y)=PrfX6x;Y6yg:(1)F(x;y),,@2F(x;y)@x@y=f(x;y);(2)F(x;y)=Zy¡1Zx¡1f(u;v)dudv:(3)2,(running).,,.62f(x;y)XY.(X;Y).(X;Y)Prfx6X6x+¢x;y6Y6y+¢yg(4)=F(x+¢x;y+¢y)¡F(x+¢x;y)¡F(x;y+¢y)+F(x;y)=Zy+¢yyZx+¢xxf(u;v)dudv;¢x0;¢y0.(X;Y)E(E)Prf(X;Y)2Eg=ZZEf(x;y)dxdy:(5)[(4)].f(x;y),f(x;y)¢y¢xXxx+¢xYyy+¢y,Prfx6X6x+¢x;y6Y6y+¢yg=Zy+¢yyZx+¢xxf(u;v)dudv(6)=f(x0;y0)¢x¢y;x0;y0(x6x06x+¢x;y6y06y+¢y).f(u;v),(6)f(x;y)¢x¢y.,lim¢x!0¢y!01¢x¢yjPrfx6X6x+¢x;y6Y6y+¢yg¡f(x;y)¢x¢yj=0:(7)pX1;X2;¢¢¢;Xp.cdfF(x1;¢¢¢;xp)=PrfX16x1;¢¢¢;Xp6xpg(8)x1;¢¢¢;xp.F(x1;¢¢¢;xp),@pF(x1;¢¢¢;xp)@x1¢¢¢@xp=f(x1;¢¢¢;xp)(9)(),F(x1;¢¢¢;xp)=Zxp¡1¢¢¢Zx1¡1f(u1;¢¢¢;up)du1¢¢¢dup:(10)p()RPrf(X1;¢¢¢;Xp)2Rg=Z¢¢¢RZf(x1;¢¢¢;xp)dx1¢¢¢dxp:(11)f(x1;¢¢¢;xp),f(x1;¢¢¢;xp)¢x1¢¢¢¢xpPrfx16X16x1+¢x1;¢¢¢;xp6Xp6xp+¢xpg.E(Xh11¢¢¢Xhpp)=Z1¡1¢¢¢Z1¡1xh11¢¢¢xhppf(x1;¢¢¢;xp)dx1¢¢¢dxp:(12)E.2.272.2.2X;YcdfF(x;y),XcdfPrfX6xg=PrfX6x;Y61g(13)=F(x;1):F(x).F(x)=Zx¡1Z1¡1f(u;v)dvdu:(14)Z1¡1f(u;v)dv=f(u)(15)X.(14)F(x)=Zx¡1f(u)du:(16)YcdfG(y)g(y)..X1;¢¢¢;XpcdfF(x1;¢¢¢;xp),X1;¢¢¢;Xp,X1;¢¢¢;Xr(rp)cdf.PrfX16x1;¢¢¢;Xr6xrg(17)=PrfX16x1;¢¢¢;Xr6xr;Xr+161;¢¢¢;Xp61g=F(x1;¢¢¢;xr;1;¢¢¢;1):X1;¢¢¢;XrZ1¡1¢¢¢Z1¡1f(x1;¢¢¢;xr;ur+1;¢¢¢;up)dur+1¢¢¢dup:(18)X1;¢¢¢;Xp.,,E(Xh11¢¢¢Xhrr)=E(Xh11¢¢¢XhrrX0r+1¢¢¢X0p)(19)=Z1¡1¢¢¢Z1¡1xh11¢¢¢xhrrf(x1;¢¢¢;xp)dx1¢¢¢dxp=Z1¡1¢¢¢Z1¡1xh11¢¢¢xhrr¢·Z1¡1¢¢¢Z1¡1f(x1;¢¢¢;xp)dxr+1¢¢¢dxp¸dx1¢¢¢dxr:2.2.3cdfF(x;y)X;Y,F(x;y)=F(x)G(y);(20)F(x)Xcdf,G(y)Ycdf.X;Y82f(x;y)=@2F(x;y)@x@y=@2F(x)G(y)@x@y(21)=dF(x)dx¢dG(y)dy=f(x)g(y):,f(x;y)=f(x)g(y),F(x;y)=Zy¡1Zx¡1f(u;v)dudv=Zy¡1Zx¡1f(u)g(v)dudv(22)=Zx¡1f(u)duZy¡1g(v)dv=F(x)G(y):,,f(x;y)=f(x)g(y).,x1x2;y1y2,Prfx16X6x2;y16Y6y2g(23)=Zy2y1Zx2x1f(u;v)dudv=Zx2x1f(u)duZy2y1g(v)dv=Prfx16X6x2gPrfy16Y6y2g:XY,XY.X1;¢¢¢;XpcdfF(x1;¢¢¢;xp),,F(x1;¢¢¢;xp)=F1(x1)¢¢¢Fp(xp);(24)Fi(xi)Xicdf,i=1;¢¢¢;p.X1;¢¢¢;XrXr+1;¢¢¢;Xp,F(x1;¢¢¢;xp)=F(x1;¢¢¢;xr;1;¢¢¢;1)¢F(1;¢¢¢;1;xr+1;¢¢¢;xp):(25).,X1;¢¢¢;Xp,E(Xh11¢¢¢Xhpp)=Z1¡1¢¢¢Z1¡1xh11¢¢¢xhppf1(x1)¢¢¢fp(xp)dx1¢¢¢dxp(26)=pYi=1Z1¡1xhiifi(xi)dxi=pYi=1fE(Xhii)g:2.2.4AB,ABP(AB),BP(B)0,BAP(AB)=P(B).2.29AX[x1;x2],BY[y1;y2].Y[y1;y2]X[x1;x2]Prfx16X6x2jy16Y6y2g=Prfx16X6x2;y16Y6y2gPrfy16Y6y2g(27)=Rx2x1Ry2y1f(u;v)dvduRy2y1g(v)dv:y1=y;y2=y+¢y.,Zy+¢yyg(v)dv=g(y¤)¢y;(28)y6y¤6y+¢y.Zy+¢yyf(u;v)dv=f[u;y¤(u)]¢y;(29)y6y¤(u)6y+¢y.,Prfx16X6x2jy6Y6y+¢yg=Zx2x1f[u;y¤(u)]g(y¤)du:(30),y¢y(0),(30).g(y)0y,Prfx16X6x2jY=yg,YyXx1x2,¢y!0(30).Prfx16X6x2jY=yg=Zx2x1f(ujy)du;(31)f(ujy)=f(u;y)=g(y).y,f(ujy),yX.,XY,f(xjy)=f(x).,X1;¢¢¢;XpcdfF(x1;¢¢¢;xp),Xr+1=xr+1;¢¢¢;Xp=xp,X1;¢¢¢;Xrf(x1;¢¢¢;xp)R1¡1¢¢¢R1¡1f(u1;¢¢¢;ur;xr+1;¢¢¢;xp)du1¢¢¢dur:(32),Chung(1974),Kolmogorov(1950),Loµeve(1977),Loµeve(1978)Neveu(1965).2.2.5X1;¢¢¢;Xpf(x1;¢¢¢;xp).pyi=yi(x1;¢¢¢;xp);i=1;¢¢¢;p:(33)xy,xi=xi(y1;¢¢¢;yp);i=1;¢¢¢;p:(34)Y1;¢¢¢;Yp,xf(x1;¢¢¢;xp).102Yi=yi(X1;¢¢¢;Xp);i=1;¢¢¢;p:(35)Y1;¢¢¢;Ypg(y1;¢¢¢;yp)=f[x1(y1;¢¢¢;yp);¢¢¢;xp(y1;¢¢¢;yp)]J(y1;¢¢¢;yp);(36)J(y1;¢¢¢;yp)J(y1;¢¢¢;yp)=mod¯¯¯¯¯¯¯¯¯¯¯@x1@y1@x1@y2¢¢¢@x1@yp@x2@y1@x2@y2¢¢¢@x2@yp.........@xp@y1@xp@y2¢¢¢@xp@yp¯¯¯¯¯¯¯¯¯¯¯:(37),mod.(X1;¢¢¢;Xp)R(11),(Y1;¢¢¢;Yp)SPrf(Y1;¢¢¢;Yp)2Sg=Z¢¢¢SZg(y1;¢¢¢;yp)dy1¢¢¢dyp:(38)SR,R(33)S,S(34)R,(11)(38).(36)Y1;¢¢¢;Yp.2.3ke¡12®(x¡¯)2=ke¡12(x¡¯)®(x¡¯);(1)®,k(1)x1.X1;¢¢¢;Xp.xx=0BB@x1...xp1CCA(2),¯b=0BB@b1...bp1CCA(3),®()A=0BBBBB@a11a12¢¢¢a1pa21a22¢¢¢a2p.........ap1ap2¢¢¢app1CCCCCA(4)2.311.®(x¡¯)2=(x¡¯)®(x¡¯)(x¡b)0A(x¡b)=pXi;j=1aij(xi¡bi)(xj¡bj)(5).pf(x1;¢¢¢;xp)=Ke¡12(x¡b)0A(x¡b);(6)K(0)x1;¢¢¢;xpp1.(6)(1)...f(x1;¢¢¢;xp).A,(x¡b)0A(x¡b)0;(7)(6),f(x1;¢¢¢;xp)6K:(8)(6)p1K.K¤=Z1¡1¢¢¢Z1¡1e¡12(x¡b)0A(x¡b)dxp¢¢¢dx1:(9)(A.1.6),A,C,C0AC=I;(10)I,C0C.x¡b=Cy;(11)y=0BB@y1...yp1CCA:(12)(x¡b)0A(x¡b)=y0C0ACy=y0y:(13)J=modjCj;(14)modjCjC.(9)K¤=modjCjZ1¡1¢¢¢Z1¡1e¡12y0ydyp¢¢¢dy1:(15)e¡12y0y=expá12pXi=1y2i!=pYi=1e¡12y2i;(16)exp(z)=ez.(15)122K¤=modjCjZ1¡1¢¢¢Z1¡1e¡12y21¢¢¢e¡12y2pdyp¢¢¢dy1(17)=modjCjpYi=1½Z1¡1e¡12y2idyi¾=modjCjpYi=1fp2g=modjCj(2)12p1p2Z1¡1e¡12t2dt=1:(18)(10)jC0j¢jAj¢jCj=jIj:(19)jC0j=jCj;(20)jIj=1,(19)modjCj=1=pjAj:(21)K=1=K¤=pjAj(2)¡12p:(22)pjAj(2)12pe¡12(x¡b)0A(x¡b):(23)X1;¢¢¢;XpbA.,X=0BB@X1...Xp1CCA:(24),,1.2.3.1ZZ11;¢¢¢;Zmn,Z=(Zgh);g=1;¢¢¢;m;h=1;¢¢¢;n:(25)Z11;¢¢¢;Zmn,Z,Z(1);¢¢¢;Z(q).Z=Z(i)pi,E(Z)Pqi=1Z(i)pi.E(Z)=(E(Zgh)).Z11;¢¢¢;Zmn,,E(Z)(),E(Z)=(E(Zgh))..2.3.2Z2.313E(Z)=(E(Zgh));g=1;¢¢¢;m;h=1;¢¢¢;n:(26),Z(24)X,E(X)=0BB@E(X1)...E(Xp)1CCA(27)X.¹.Z(X¡¹)(X¡¹)0,C(X)=E(X¡¹)(X¡¹)0=[E(Xi¡¹i)(Xj¡¹j)];(28)X.iE(Xi¡¹i)2Xi,i;jE(Xi¡¹i)(Xj¡¹j)XiXj,i6=j.§.C(X)=E(XX0¡¹X0¡X¹0+¹¹0)=E(XX0)¡¹¹0:(29)(),.2.3.1Zm£n,Dl£m,En£q,Fl£q,E(DZE+F)=D(E(Z))E+F:(30)E(DZE+F)ijE0@Xh;gdihZhgegj+fij1A=Xh;gdih(E(Zhg))egj+fij;(31)D(E(Z))E+Fij.¥2.3.2Y=DX+f,X,E(Y)=DE(X)+f;(32)C(Y)=DC(X)D0:(33)2.3.1,,C(Y)=E(Y¡E(Y))(Y¡E(Y))0(34)=E[DX+f¡(DE(X)+f)][DX+f¡(DE(X)+f)]0