The problem of comparing two (or more samples) appears in several and diverse applications. The parametric theory resolves the problem by appealing to the well known t-test. To carry out the t-test both samples are assumed to be normally distributed with common unknown variance and unknown means. The two sample t-test enjoys several optimality properties, for instance it is uniformly most powerful unbiased test. Occasionally some (or all) of the needed assumptions fail; for instance, when there exists a group of observations with skewed distribution, then both assumptions of normality and equality of variances do not hold true. Hence, application of the ordinary two-sample t-test is questionable. The problem is usually bypassed after a suitable transformation but the comparison need to be carried out in the transformed scale. Alternatively, we can appeal to the nonparametric theory that approaches the problem of comparing two samples by the so-called Mann–Whitney–Wilcoxon test (see Wilcoxon–Mann–Whitney...
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Fokianos, K. (2011). Density Ratio Model. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_206
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