ODC and ROC Curves, Comparison Curves and Stochastic Dominance

We discuss two novel approaches to inter-distributional comparisons in the classical two-sample problem. Our starting point is properly standardised 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 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 summarise 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 that 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.