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We discuss two novel approaches to the classical two-sample problem. Our
starting point are properly standardized and combined, very popular in several
areas of statistics and data analysis, ordinal dominance and receiver
characteristic curves, denoted by ODC and ROC, respectively. The proposed new
curves are termed the comparison curves. Their estimates, being weighted rank
processes on (0,1), form the basis of inference. These weighted processes are
intuitive, well-suited for visual inspection of data at hand, and are also
useful for constructing some formal inferential procedures. They can be applied
to several variants of two-sample problem. Their use can help to improve some
existing procedures both in terms of power and the ability to identify the
sources of departures from the postulated model. To simplify interpretation of
finite sample results we restrict attention to values of the processes on a
finite grid of points. This results in the so-called bar plots (B-plots) which
readably summarize the information contained in the data. What is more, we show
that B-plots along with adjusted simultaneous acceptance regions provide
principled information about where the model departs from the data. This leads
to a framework which facilitates identification of regions with locally
significant differences.

We show an implementation of the considered techniques to a standard
stochastic dominance testing problem. Some min-type statistics are introduced
and investigated. A simulation study compares two tests pertinent to the
comparison curves to well-established tests in the literature and demonstrates
the strong and competitive performance of the former in many typical
situations. Some real data applications illustrate simplicity and practical
usefulness of the proposed approaches. A range of other applications of
considered weighted processes is briefly discussed too.