第三章多元正态总体参数的假设检验

1、3.2 例3.2.1x=3.7 48.5 9.3;5.7 65.1 8;3.8 47.2 10.9;3.2 53.2 12;3.1 55.5 9.7; 4.6 36.1 7.9;2.4 24.8 14;7.2 33.1 7.6;6.7 47.4 8.5;5.4 54.1 11.3;3.9 36.9 12.7;4.5 58.8 12.3;3.5 27.8 9.8;4.5 40.2 8.4;1.5 13.5 10.1;8.5 56.4 7.1;4.5 71.6 8.2;6.5 52.8 10.9;4.1 44.1 11.2;5.5 40.9 9.4u0=4 50 10;n=20;T2=n*(n-1

2、)*(mean(x)-u0)*inv(19*cov(x)*(mean(x)-u0),p=3;F=(n-p)*T2/(n-1)*p),p=1-fcdf(F,3,17)T2 = 9.7388F = 2.9045p =0.0649在显著性水平0.05下，接受原假设。第二类错误的计算，用非中心的F分布计算，非中心的参数为p67页，中间的参数。ncfcdf(3.2,3,17,20*(mean(x)-4 50 10)*inv(cov(x)*(mean(x)-4 50 10)ans = 0.391例3.2.2 x=3.7 48.5 9.3;5.7 65.1 8;3.8 47.2 10.9;3.2 53.2

3、12;3.1 55.5 9.7; 4.6 36.1 7.9;2.4 24.8 14;7.2 33.1 7.6;6.7 47.4 8.5;5.4 54.1 11.3;3.9 36.9 12.7;4.5 58.8 12.3;3.5 27.8 9.8;4.5 40.2 8.4;1.5 13.5 10.1;8.5 56.4 7.1;4.5 71.6 8.2;6.5 52.8 10.9;4.1 44.1 11.2;5.5 40.9 9.4S=cov(x),v,d=eig(S),n=20;p=3;c2=(n-1)*p*3.2/(n*(n-p),d123=diag(sqrt(d)*sqrt(c2)S = 2

4、.8794 10.0100 -1.8091 10.0100 199.7884 -5.6400 -1.8091 -5.6400 3.6277v = -0.8175 0.5737 0.0508 0.0249 -0.0530 0.9983 -0.5754 -0.8173 -0.0291d = 1.3014 0 0 0 4.5316 0 0 0 200.4625c2 = 0.5365d123 = 0.8356 1.5592 10.37032、联立置信区间由3.2.4式计算的T2区间为：x=3.7 48.5 9.3;5.7 65.1 8;3.8 47.2 10.9;3.2 53.2 12;3.1 55.

5、5 9.7; 4.6 36.1 7.9;2.4 24.8 14;7.2 33.1 7.6;6.7 47.4 8.5;5.4 54.1 11.3;3.9 36.9 12.7;4.5 58.8 12.3;3.5 27.8 9.8;4.5 40.2 8.4;1.5 13.5 10.1;8.5 56.4 7.1;4.5 71.6 8.2;6.5 52.8 10.9;4.1 44.1 11.2;5.5 40.9 9.4S=cov(x),v,d=eig(S),n=20;p=3;c2=(n-1)*p*3.2/(n*(n-p),d123=diag(sqrt(d)*sqrt(c2) mean(x)-sqrt(n

6、-1)*p*3.2/(n-p)*sqrt(diag(S)/n) mean(x)+sqrt(n-1)*p*3.2/(n-p)*sqrt(diag(S)/n)ans = 3.3971 5.8829 35.0472 55.75288.5700 11.3600按3.2.5计算的区间为： mean(x)-tinv(0.975,n-1)*sqrt(diag(S)/n) mean(x)+tinv(0.975,n-1)*sqrt(diag(S)/n)ans = 3.8458 5.4342 38.7848 52.0152 9.0736 10.85643.3例3.3.1一、假定两总体方差相同x=65 35 25

7、60;75 50 20 55;60 45 35 65;75 40 40 70;70 30 30 50;55 40 35 65;60 45 30 60;65 40 25 60;60 50 30 70;55 55 35 75y=55 55 40 65;50 60 45 70;45 45 35 75;50 50 50 70;55 50 30 75;60 40 45 60;65 55 45 75;50 60 35 80;40 45 30 65;45 50 45 70n=10;m=10;p=4;mx=mean(x),my=mean(y),A1=(n-1)*cov(x),A2=(m-1)*cov(y),D

8、2=(n+m-2)*(mx-my)*inv(A1+A2)*(mx-my),T2=n*m*D2/(m+n),F=(n+m-p-1)*T2/(n+m-2)*p),p=1-fcdf(F,4,15)mx = 64.0000 43.0000 30.5000 63.0000my = 51.5000 51.0000 40.0000 70.5000A1 = 490.0000 -170.0000 -120.0000 -245.0000 -170.0000 510.0000 10.0000 310.0000 -120.0000 10.0000 322.5000 260.0000 -245.0000 310.000

9、0 260.0000 510.0000A2 = 502.5000 60.0000 175.0000 -7.5000 60.0000 390.0000 50.0000 195.0000 175.0000 50.0000 450.0000 -100.0000 -7.5000 195.0000 -100.0000 322.5000D2 = 5.9725T2 = 29.8625F = 6.2214p = 0.0037二、假定两总体方差不相同z=x-y;n=10;p=4,T2=(n-1)*n*mean(z)*inv(n*cov(z,1)*mean(z),F=(n-p)*T2/(n-1)*p),p=1-f

10、cdf(F,p,n-p)p = 4T2 = 31.5536F = 5.2589p = 0.0364在显著性水平0.05下拒绝原假设。三、当n不等于m时若第9、10行没有，则nF 4.7592e-004 stats = gnames: 3x1 char n: 20 20 20 source: anova1 means: 2.0000e+002 2.0000e+002 2.0000e+002 df: 57 s: 46.659注意：第86页，求D的算法：如协差阵为：a=2 -2;-2 9v,d=eig(a)v = -0.467 -0.702 -0.702 0.467d = 1.073 0 0 9.9

11、27c=(v*sqrt(inv(d),c*a*cc = -0.586 -0.067 -0.469 0.160ans = 1.000 -0.000 -0.000 1.000注意：cov(x*d)=d*cov(x)*d例3.4.1p=4;k=3;n=60;n1=20;A1=n1*cov(xa,1);A2=n1*cov(xb,1);A3=n1*cov(xc,1);A=A1+A2+A3;d=(2*p2+3*p-1)*(k+1)/(6*(p+1)*(n-k),M=(n-k)*log(det(A/(n-k)-(n1-1)*(log(det(A1/(n1-1)+log(det(A2/(n1-1)+log(d

12、et(A3/(n1-1),p=1-chi2cdf(1-d)*M,0.5*p*(p+1)*(k-1)d = 0.1006M = 22.6054p =0.4374例3.5.1x=3.7 48.5 9.3;5.7 65.1 8;3.8 47.2 10.9;3.2 53.2 12;3.1 55.5 9.7; 4.6 36.1 7.9;2.4 24.8 14;7.2 33.1 7.6;6.7 47.4 8.5;5.4 54.1 11.3;3.9 36.9 12.7;4.5 58.8 12.3;3.5 27.8 9.8;4.5 40.2 8.4;1.5 13.5 10.1;8.5 56.4 7.1;4.5

13、 71.6 8.2;6.5 52.8 10.9;4.1 44.1 11.2;5.5 40.9 9.4n=20;p=3;p1=1;b=n-1.5-(1-3)/(3*(1-3),f=0.5*(p*(p+1)-3*2),V=det(20*cov(x,1)/prod(diag(20*cov(x,1), p=1-chi2cdf(-b*log(V),f)%prod 连乘b = 18.1667f = 3V = 0.5665p = 0.01603.6QQ图x=normrnd(2,4,100,1);y=sort(x);t=1:100;k=norminv(t-0.5)/100);scatter(k,y)hold

14、onplot(-3,3,-10,14,:)直线为；y=标准差x+u散点图为：它与qqplot(x)相同qqplot(x)x=normrnd(2,4,100,1); y=sort(x); t=1:100;k=norminv(t-0.5)/100,2,4)scatter(y,k) hold onplot(-10,15,-10,15) Q-Q图PP图plot(t-0.5)/100,normcdf(y,2,4),bo,markersize,10)hold on plot(0,1,0,1,r:,linewidth,5) P-P图P102 最后一题xa=260 75 40 18;200 72 34 17;

15、240 87 45 18;170 65 39 17;270 110 39 24;205 130 34 23;190 69 27 15;200 46 45 15;250 117 21 20;200 107 28 20;225 130 36 11;210 125 26 17;170 64 31 14;270 76 33 13;190 60 34 16;280 81 20 18;310 119 25 15;270 57 31 8;250 67 31 14;260 135 39 29mxa=mean(xa);D2=diag(xa-repmat(mxa,1,20)*inv(cov(xa)*(xa-rep

16、mat(mxa,1,20)y=sort(D2); t=1:20;k=chi2inv(t-0.5)/20,4)scatter(y,k) hold onplot(0,12,0,12) QQD2 = 2.3566 0.8757 3.3048 2.8115 3.7483 3.4172 2.5699 4.3461 3.5919 2.1021 11.0806 3.6420 3.0201 2.0029 1.7946 6.3343 5.2383 5.1832 1.2706 7.3091注意D2的也可这样计算：D2=zeros(20,1);for i=1:20;D2(i,1)=(xa(i,:)-mxa)*inv(cov(xa)*