Shapiro-Wilk vs Anderson Darling test for normality What makes the Shapiro-Wilk test have the best power for a given significance when compared to Anderson-Darling test and other tests.

Wikipedia on 'Normality tests' says [among frequentist (non-Bayesian) tests] "A 2011 study concludes that Shapiro–Wilk has the best power for a given significance, followed closely by Anderson–Darling when comparing the Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors, and Anderson–Darling tests." and provides the reference (w/ PDF). Perhaps the reference only gives results of dozens of simulations, and perhaps it answers your question more directly. That seems a good place to start.
I would be surprised if one cannot find any specific non-normal distribution for which A-D has higher power.
