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Data-driven most powerful tests are statistical hypothesis decision-makingtools that deliver the greatest power against a fixed null hypothesis among allcorresponding data-based tests of a given size. When the underlying datadistributions are known, the likelihood ratio principle can be applied toconduct most powerful tests. Reversing this notion, we consider the followingquestions. (a) Assuming a test statistic, say T, is given, how can we transformT to improve the power of the test? (b) Can T be used to generate the mostpowerful test? (c) How does one compare test statistics with respect to anattribute of the desired most powerful decision-making procedure? To examinethese questions, we propose one-to-one mapping of the term 'Most Powerful' tothe distribution properties of a given test statistic via matchingcharacterization. This form of characterization has practical applicability andaligns well with the general principle of sufficiency. Findings indicate thatto improve a given test, we can employ relevant ancillary statistics that donot have changes in their distributions with respect to tested hypotheses. Asan example, the present method is illustrated by modifying the usual t-testunder nonparametric settings. Numerical studies based on generated data and areal-data set confirm that the proposed approach can be useful in practice.

A Characterization of Most(More) Powerful Test Statistics with Simple Nonparametric Applications