Nonparametric weighted symmetry tests

Weighted symmetry is an extension of the classical notion of symmetry in which the tails of a distribution are similar, up to a scaling factor. The authors develop test statistics of weighted symmetry based on empirical processes. The finite-dimensional distributions of the proposed statistics are either nonparametric or conditionally nonparametric, according as the parameters of weighted symmetry are known or estimated. Asymptotically, the distributions of the processes behave like Brownian bridges or motions, leading to familiar distributions for the proposed test statistics. The authors also establish the asymptotic normality of Hodges-Lehmann type estimators based on a generalization of the Wilcoxon signed rank test. Furthermore, they propose density estimators in that setting.