The problem addressed is that of finding the "closest" symmetric distribution (density) to a given theoretical or empirical distribution (density) function. Measures of "closeness" considered include: weighted sup norm, weighted $L_p$ norm and Hellinger distance. Explicit formulas are given for the closest symmetric distribution function to the empirical distribution function in both sup norm and integrated square error.
"Identifying the Closest Symmetric Distribution or Density Function." Ann. Statist. 15 (2) 865 - 874, June, 1987. https://doi.org/10.1214/aos/1176350380