Geodesic
Two families of normality tests based on Polya characterization, and their efficiency
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 147-159

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For testing of normality we introduce two families of statistics based on extended Polya characterization of the normal law. The first family depends on parameter $a\in(0,1)$, and for any $a$ its members are asymptotically normal and consistent for many alternatives of interest. We study the local Bahadur efficiency of these statistics as a function of $a$ and find that for common alternatives the Polya case $a=1/\sqrt{2}$ is the worst and the maximum of efficiency is attained for $a$ close to 0 or 1. The second family depends on natural $m$ and the efficiency increases when $m$ grows. 
@article{ZNSL_2005_328_a7,
     author = {V. V. Litvinova and Ya. Yu. Nikitin},
     title = {Two families of normality tests based on {Polya} characterization, and their efficiency},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {147--159},
     publisher = {mathdoc},
     volume = {328},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_328_a7/}
}
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V. V. Litvinova; Ya. Yu. Nikitin. Two families of normality tests based on Polya characterization, and their efficiency. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 147-159. http://geodesic.mathdoc.fr/item/ZNSL_2005_328_a7/