We discuss two-sample problems and the implementation of a new two-sample dataanalysis procedure. The proposed procedure is based on the concepts of mid-distribution,design of score functions, components, comparison distribution, comparison densityand exponential model. Assume that we have a random sample X1, . . . ,Xm froma continuous distribution F(y) = P(Xi y), i = 1, . . . ,m and a random sampleY1, . . . ,Yn from a continuous distribution G(y) = P(Yi y), i = 1, . . . ,n. Also assumeindependence of the two samples. The two-sample problem tests homogeneity oftwo samples and formally can be stated as H0 : F = G. To solve the two-sample problem,a number of tests have been proposed by statisticians in various contexts. Twotypical tests are the two-sample t?test and the Wilcoxon's rank sum test. However,since they are testing differences in locations, they do not extract more informationfrom the data as well as a test of the homogeneity of the distribution functions. Eventhough the Kolmogorov-Smirnov test statistic or Anderson-Darling tests can be usedfor the test of H0 : F = G, those statistics give no indication of the actual relationof F to G when H0 : F = G is rejected. Our goal is to learn why it was rejected.Our approach gives an answer using graphical tools which is a main property of ourapproach. Our approach is functional in the sense that the parameters to be estimatedare probability density functions. Compared with other statistical tools fortwo-sample problems such as the t-test or the Wilcoxon rank-sum test, density estimation makes us understand the data more fully, which is essential in data analysis.Our approach to density estimation works with small sample sizes, too. Also ourmethodology makes almost no assumptions on two continuous distributions F andG. In that sense, our approach is nonparametric. Our approach gives graphical elementsin two-sample problem where exist not many graphical elements typically.Furthermore, our procedure will help researchers to make a conclusion as to why twopopulations are different when H0 is rejected and to give an explanation to describethe relation between F and G in a graphical way.
