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

Asymptotic Efficiency of Goodness-of-fit Tests for the Power Function Distribution Based on Puri--Rubin Characterization