  We propose two families of scale-free exponentiality tests based on the
recent characterization of exponentiality by Arnold and Villasenor. The test
statistics are based on suitable functionals of U-empirical distribution
functions. The family of integral statistics can be reduced to V- or
U-statistics with relatively simple non-degenerate kernels. They are
asymptotically normal and have reasonably high local Bahadur efficiency under
common alternatives. This efficiency is compared with simulated powers of new
tests. On the other hand, the Kolmogorov type tests demonstrate very low local
Bahadur efficiency and rather moderate power for common alternatives,and can
hardly be recommended to practitioners. We also explore the conditions of local
asymptotic optimality of new tests and describe for both families special "most
favorable" alternatives for which the tests are fully efficient.
