On Bahadur efficiency in goodness of fit testing: A Review of recent results and challenges

Abstract

Although introduced in the 1960s, recently the Bahadur efficiency becomes a popular tool for accessing the quality of goodness-of-fit tests. The reason for it lay in the fact that it reflects the test behavior for large sample sizes. In addition, unlike some other types of asymptotic efficiencies, it is suitable for tests with non-normal limiting distributions. It turns out that the essential ingredient for the calculation of the Bahadur efficiency is the so-called large deviation function which is, in general, not easy to obtain. In that case, as an alternative approach, the notion of approximate Bahadur efficiency might be used. It proves to be a good approximation of the exact Bahadur efficiency of tests against close alternatives. In this talk, we outline the Bahadur theory and illustrate its application on a few classical and recently proposed goodness-of-fit tests. A significant part of the talk will be dedicated to current challenges in calculations of Bahadur efficiency and potential directions for future research.