  We study the Bahadur efficiency of several weighted L2--type
goodness--of--fit tests based on the empirical characteristic function. The
methods considered are for normality and exponentiality testing, and for
testing goodness--of--fit to the logistic distribution. Our results are helpful
in deciding which specific test a potential practitioner should apply. For the
celebrated BHEP and energy tests for normality we obtain novel efficiency
results, with some of them in the multivariate case, while in the case of the
logistic distribution this is the first time that efficiencies are computed for
any composite goodness--of--fit test.
