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The two-sample problem usually tests for a difference in location. However, there are many situations, for example in biomedicine, where jointly testing for difference in location and variability may be more appropriate. Moreover, heavy-tailed data, outliers and small-sample sizes are common in biomedicine and in other fields. These considerations make the use of nonparametric methods more appealing than parametric ones. The aim of the paper is to contribute to the literature about nonparametric simultaneous location and scale testing. More precisely, several existing tests are generalized and unified, and a new class of tests based on the Mahalanobis distance between the percentile modified test statistics for location and scale differences is introduced. The asymptotic distributions of the test statistics are obtained, and small-sample size behaviour of the tests is studied and compared to other tests via Monte Carlo simulations. It is shown that the proposed class of tests performs well when there are differences in both location and variability. A practical application is presented.

The main objective of this thesis is the presentation regarding the problem of testing independence between two random variables in the nonparametric model of continuous cumulative distribution functions. Firstly, the reader is informed with basic notions from the theory of independence and rank tests. Afterwards, few of the most common methods for testing independence are introduced. In the beginning, the test based on Pearson's correlation coefficient is mentioned as a representative for parametric tests, then we continue with nonparametric tests, such as test based on Spearman's, Kendall's and distance correlation coefficient. We focus in better detail on Hoeffding's test of independence, which results to be consistent against all alternatives in the model of continuous cumulative distribution functions. In the end, we compare and evaluate presented methods for testing independence using simulations in R environment.