Asymptotic power of tests of normality under local alternatives

Seven tests of univariate normality are studied in view of their asymptotic power under local alternatives. The procedures under consideration are either based on the empirical skewness and/or kurtosis, including the popular Jarque–Bera statistic, as well as Cramér–von Mises, Anderson–Darling and Kolmogorov–Smirnov functionals of an empirical process with estimated parameters. The large-sample behavior of these test statistics under contiguous sequences is obtained; this allows for the computation of their associated local power curves and of their asymptotic relative efficiency in the light of a measure proposed by Berg and Quessy (2009). Comparisons are made under four classes of local alternatives, including those used by Thadewald and Büning (2007) in a recent Monte-Carlo power study. These theoretical results are related to empirical ones and many recommendations are formulated.