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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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

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