A new asymptotic distribution for Hollander's bivariate symmetry statistic. (Q1566658)

We show that the asymptotic distribution [\textit{J. A. Koziol}, Commun. Stat., Theory Methods A8, 207--221 (1979; Zbl 0445.62063)] of the bivariate symmetry statistic standardized by \(1/n^{3}\), where \(n\) is the number of iid paired observations from bivariate distribution F, fails to account for the conditional nature of the statistic and is very conservative for all n. We describe the dependence of the true critical values of the test on the covariance of the paired samples, \(\sigma_{xy}\), which is a nuisance parameter in this problem. We propose an alternative standardization of the bivariate symmetry statistic that adjusts it for \(\sigma_{xy}\) as well as n. We show via a simulation study that the size of the proposed asymptotic test is close to the nominal level, even for small n.