Goodness-of-fit tests based on the characterization of uniformity by the ratio of order statistics, and their efficiencies

We construct integral and supremum type goodness-of-fit tests for the uniform law based on Ahsanullah's characterization of uniform law. We discuss limiting distributions of new tests and describe the logarithmic large deviation asymptotics of test statistics under null-hypothesis. This enables to calculate their local Bahadur efficiency under some parametric alternatives. Conditions of local optimality of new statistics are given.