A test for normality based on the empirical characteristic function

SUMMARY An omnibus test of normality is proposed, which has high power against many alternative hypotheses. The test uses a weighted integral of the squared modulus of the difference between the characteristic functions of the sample and of the normal distribution. together with the one-to-one correspondence between +(t) and F(x), suggest utilizing kn(t) in statistical inference. Here we use /n(t) to test the composite hypothesis that F(x) is normal. Let 00(t) = exp (it1 - 1t2 a2) be the characteristic function under the null hypothesis, with 1u and U2 the unspecified mean and variance. The test introduced here is based on a weighted integral over t of the squared modulus of kn(t) - $O(t), where $0(t) depends on sample estimates of ju and a2. We compare the power of the test with that of prominent tests based on order statistics and on sample moments. Heathcote (1972) and Feigin & Heathcote (1977) have previously considered using either the real or imaginary part alone of 4n(t) in tests of simple hypotheses. Their tests relied on the fact that, for given t, either component of 4n(t) is asymptotically normal with mean given by its population counterpart. If the alternative hypothesis is also simple, it may be possible to find a value of t that maximizes the power of the test in large samples. Epps, Singleton & Pulley (1982) showed how the sample moment generating function may be employed to test composite hypotheses that the data come from one or the other of two separate families of distributions. Murota & Takeuchi (1981) have recently proposed a location and scale invariant test based on the statistic -an(t) = I n(t/S) 12, where S is the sample standard deviation. Applied to the problem of testing normality, an(t) was found to have high power in the vicinity of t = 10 against members of six families of symmetric alternatives, but its