Testing Gaussianity with the characteristic function: The iid case

Recently, tests for Gaussianity have been proposed which use the characteristic function. It is the purpose of this paper to highlight and resolve some problems with these tests and to improve performance so that the characteristic function based test is competitive with, and in some cases better than, the most powerful known tests for Gaussianity. We will focus our studies upon the relevant case where only a small amount of data is available. It is in this case when the power of the test one uses becomes most critical; it is therefore unfortunate that when only a small amount of data is available the sampling distributions of known Gaussianity test statistics break down. We relate our experimental observations for small sample sizes (less than one hundred points) where essentially all such test statistic distributions should be viewed with great suspicion. Thus, in these studies, for lack of any acceptable theoretical results, we replace such distributional theories with empirically derived thresholds, which are guaranteed to be accurate but with an associated computational cost.