  In this paper some well-known tests based on empirical distribution functions
(EDF) with estimated parameters for testing composite normality hypothesis are
revisited, and some new results on asymptotic properties are provided. In
particular, the approximate Bahadur slopes are obtained -- in the case of close
alternatives -- for the EDF-based tests as well as the likelihood ratio test.
The local approximate efficiencies are calculated for several close
alternatives. The obtained results could serve as a benchmark for evaluation of
the quality of recent and future normality tests.
