Two families of normality tests based on Polya characterization, and their efficiency

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.