Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter

We generalize signed rank statistics to dimensions higher than one. This results in a class of orthogonally invariant and distribution free tests that can be used for testing spherical symmetry/location parameter. The corresponding estimator is orthogonally equivariant. Both the test and estimator can be chosen with asymptotic efficiency 1. The breakdown point of the estimator depends only on the scores, not on the dimension of the data. For elliptical distributions, we obtain an affine invariant test with the same asymptotic properties, if the signed rank statistic is applied to standardized data. We also present a method for computing the estimator numerically, and consider a real data example and some simulations. Finally, an application to detection of time-varying signals in spherically symmetric noise is given.