1、6 / 6非參數檢驗及matlab實現Kolmogorov-Smirnovtest檢驗兩個樣本是否有相同分布KstestTeststatistics:maxF(x)G(x)h,p,ksstat,cv=kstest(x,CDF,alpha,type)x：被測試的數據樣本，以列向量輸入(continuousdistributiondefinedbycumulativedistributionfunction)CDF：被檢驗的樣本cumulativedistributionfunction,缺省值為N(0,1)Alpha：顯著性水平，缺省時為0.05Type：字符輸入。unequal'(缺省

2、值)檢驗兩者分布是否相同'larger'檢驗x的CDF大于給定的CDF'smaller'檢驗x的CDF小于給定的CDFhh=0不拒絕原假設，即兩個分布相同h=1拒絕原假設，即兩個分布不同p:拒絕原假設的最小顯著性水平ksstat：假設為真時，滿足student分布cv:criticalvalue/cutoffvalue,determiningifksstatissignificant.Kstest2:h,p,ks2stat=kstest2(x1,x2,alpha,type)詳見ketestLillieforstest檢驗兩個樣本是否有相同分布Teststatis

3、tics:maxF(x)G(x)x2-sidedgoodness-of-fittestlillietesth,p,kstat,critval=lillietest(x,alpha,distr,mctol)各參數參見kstest,特別的，mctol為使用蒙特卡洛方法計算p值Jarque-Beratest檢驗樣本是否來自均值和方差未知的正態分布two-sidedgoodness-of-fittest假設：x為正態分布2teststatistic：jb-s2k3,wherenisthesamplesize,sis64thesampleskewness,andkisthesamplekurtosis.

4、jbtesth,p,jbstat,critval=jbtest(x,alpha,mctol)各個參數意義詳見lillietest。Wilcoxon-Mann-WhitneyRanksTest檢驗兩個樣本是否來自于同一分布。以下functionf=Wilcoxon_Rank_Test(x,y,alpha)直接是秩和檢驗方法，另有functionf=Pre_Wilcoxon(x,y,alpha)是改進的秩和檢驗方法。區別在于樣本的處理方式。functionf=Wilcoxon_Rank_Test(x,y,alpha);%x,yarevectors%x,ycanbeatdifferentlength

5、Z=x;y;%z1forrankingfromsmalltolarge%z2forplacez1,z2=sort(Z,'ascend');fori=1:size(x,1)forj=i:size(z1,1)ifz1(j)=x(i)X(i,1)=j;endendend%Xistherankofxinvectorm=size(x,1);n=size(y,1);Ex=m*(m+n+1)/2;Varx=m*n*(m+n+1)/12;c=sqrt(Varx)*norminv(1-alpha/2);beta=sum(X,1);ifabs(beta-Ex)>cdisp('Reje

6、ctNullHypothesis');f=1;elsedisp('FailtoRejectNullHypothesis');f=0;endfunctionf=Pre_Wilcoxon(x,y,alpha);%tofindwhetherdivisionisneededmemo=zeros(size(x,1),size(y,1);fori=1:size(x,1)forj=1:size(y,1)ifx(i,1)=y(j,1)memo(i,j)=1;endendend%memocontainstheinformationofdivision;%memoisasymmetricm

7、atrix;ifmemo=zeros(size(memo)disp('Noneedfordivision');disp('GoonRankTeststraightly');disp('ReaultasFollowing','significantlevel',num2str(alpha);Wilcoxon_Rank_Test(x,y,alpha);elsedisp('Divisionisneeded');cx,cy=find(memo>0);if(cx(1,1)=1)&&(cy(1,1)=1)

8、disp('group1');gamma=Wilcoxon_Rank_Test(x(1:cx(1),y(1:cy(1),alpha);ifgamma=1;returnendfork=1:size(cx,1)disp('group',num2str(k+1);gamma=Wilcoxon_Rank_Test(x(cx(k):cx(k+1),y(cy(k):cy(k+1),alpha);ifgamma=1;breakendendelseif(cx(1)=1)&&(cy(1)=1)|(cx(1)=1)&&(cy(1)=1)gamma=0

9、;disp('RejectNullHypothesis');end%(cx(1)=1)&&(cy(1)=1)fork=1:size(cx,1)disp('group',num2str(k+1);gamma=Wilcoxon_Rank_Test(x(cx(k):cx(k+1),y(cy(k):cy(k+1),alpha);ifgamma=1;breakendendendend另外matlab中也有相關的自帶的秩和檢驗函數：signrank/signtestp=signrank(x,y)為雙側檢驗，檢驗x,y是否來自中位數為0的整體;相似的，有p=s

10、igntest(x,y)ranksump=ranksum(x,y)為雙側檢驗，檢驗x,y之間是否相互獨立，要求x,y之間有相同的中位數Chi-squaregoodness-of-fittest離散態的檢驗樣本是否來自于已知分布。2NOiEi2Teststatistics:,whereOiaretheobservedcountsi1EiandEiaretheexpectedcounts.Thestatistichasanapproximatechisquaredistributionwhenthecountsaresufficientlylarge.chi2gofh=chi2gof(x)H0:x

11、(列向量)來自正態分布(期望和方差來源于x)(defaultsignificancelevel:0.05)h=0failtorejectH0h=1rejectH0或者h,p=chi2gof(x,'cdf',normcdf,mean(x),std(x)關于樣本是否來自對稱的分布functionf=symmetric_test(x,alpha)%xisavector%alphaisthesignificantlevelx1,x2=sort(x,'ascend');%x1isxfrommintomax%x2istheplacen=size(x,1);x_ME=median(x,1);%medianincolumnx_IQR=iqr(x);%