Comments: A review of testing procedures based on the empirical characteristic function

摘要

Professor Meintanis is to be congratulated for this nice review paper on tests based on the empirical characteristic function (ECF). There is not much to add, but I want to emphasize one basic point. There are numerous results on the limit behavior of ECF based tests under contiguous alternatives to the null hypothesis, and usually authors derive an almost sure limit for the suitably scaled test statistic under fixed alternatives. At least to my knowledge, however, there is only one result on the limit distribution of the test statistic under fixed alternatives, which is due to Gurtler (2000), and which refers to the BHEP tests for multivariate normality. Since the approach taken by Gurtler will probably be useful also for other tests based on the ECF, we outline the reasoning in what follows.