Goodness-of-fit tests for the power function distribution based on Puri–Rubin characterization and their efficiencies

We construct integral and supremum type goodness-of-fit tests for the family of power distribution functions. Test statistics are functionals of $U$-empirical processes and are based on the classical characterization of power function distribution family belonging to Puri and Rubin. We describe the logarithmic large deviation asymptotics of test statistics under null-hypothesis, and calculate their local Bahadur efficiency under common parametric alternatives. Conditions of local optimality of new statistics are given.