  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.
