  The test for normality suggested by Epps and Pulley (1983) is a serious
competitor to tests based on the empirical distribution function. In contrast
to the latter procedures, it has been generalized to obtain a genuine affine
invariant and universally consistent test for normality in any dimension. We
obtain approximate Bahadur efficiencies for the test of Epps and Pulley, thus
complementing recent results of Milo\v{s}evi\'c et al. (2021). For certain
values of a tuning parameter that is inherent in the Epps--Pulley test, this
test outperforms each of its competitors considered in Milo\v{s}evi\'c et al.
(2021), over the whole range of six close alternatives to normality.
