Intermediate efficiency of some weighted goodness-of-fit statistics

This paper introduces and investigates powerful omnibus test for uniformity and compares it with some classical and recent solutions. All statistics under consideration are weighted functionals of the classical empirical process. The goal is to provide a quantitative comparison of tests under consideration and to study real possibilities of using them to detect departures from the hypothesised distribution that occur in the tails. The newly introduced concept of pathwise variant of intermediate efficiency serves to achieve the goal. This contribution covers the case when under the alternative a moderately large portion of probability mass is allocated towards the tails. It is demonstrated that the approach allows for tractable, analytic comparison between the given test and the benchmark, and for reliable quantitative evaluation of weighted statistics. Finite sample results illustrate the proposed approach and confirm the theoretical findings.