Anil K. Bera, Antonio F. Galvao, Liang Wang, Zhijie Xiao, A NEW CHARACTERIZATION OF THE NORMAL DISTRIBUTION AND TEST FOR NORMALITY, Econometric Theory, Vol. 32, No. 5 (October 2016), pp. 1216-1252 ...

Results are given of an empirical sampling study of the sensitivities of nine statistical procedures for evaluating the normality of a complete sample. The nine statistics are W (Shapiro and Wilk, ...

These tests often employ advanced metrics, such as weighted L2-statistics and comparisons via the moment generating function, to sensitively detect departures from normality.