Måns Thulin: Directed tests for normality

We discuss the use of tests for normality, asking when, if ever, such tests should be used. We then argue that popular tests such as the Jarque-Bera and Shapiro-Wilk tests should be replaced by tests that are directed towards some class of alternative distributions. Using canonical correlations between sets of sample moments, we generalize the Lin-Mudholkar test for normality and propose tests for uni- and multivariate normality directed towards either skew or kurtotic alternatives.