The paper deals with problems of rank tests of symmetry when samples are drawn from purely discrete distributions so that ties of zero and non-zero observations may occur. Zero observations are considered in the same way as nonzero ones. Two ways of treatment of ties are used in the paper, randomization of ties and the method of averaged scores. The asymptotic distributions of the statistics are derived under hypothesis of symmetry and under contiguous alternatives of location. The asymptotic power and efficiency of tests are established.
"Asymptotic Properties of Rank Tests of Symmetry Under Discrete Distributions." Ann. Math. Statist. 43 (6) 2013 - 2018, December, 1972. https://doi.org/10.1214/aoms/1177690875