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Construction of an asymptotically distribution free test for the hypothesis that two multivariate random samples are identically distributed has been a topic among many statisticians for a long time. Although this problem has been solved for random samples of multivariate normal data within the parametric setting, there are not many studies in the literature for treating this problem with random samples from arbitrary unknown distributions. This book sheds a new light on this topic proposing few innovative nonparametric procedures which can be applied for any two random samples from unknown distributions. In our first approach we propose to establish a multiple direction rank statistic developed based on the projected data towards some arbitrary directions. Next we develop the test statistic in terms of this multiple direction rank statistic, which can be used to test whether the two samples have the same underlying distribution or not. Secondly, alternative approaches to a slightly different problem are explored. These alternative approaches are developed on the basis of paired comparisions.
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