  Two new tests for exponentiality, of integral and Kolmogorov type, are
proposed. They are based on a recent characterization and formed using
appropriate V-statistics. Their asymptotic properties are examined and their
local Bahadur efficiencies against some common alternatives are found. A class
of locally optimal alternatives for each test is obtained. The powers of these
tests, for some small sample sizes, are compared with different exponentiality
tests.
