On multivariate signed rank tests

A class of affine-invariant signed rank test statistics is considered for the one-sample multivariaie location problem. This class is obtained by modifying Randles' sign test which uses the transformation-retransformation approach of Chakraborty, Chaudhuri and Oja along with a directional transformation due to Tyler. It contains test statistics that have been previously studied for this problem. For the class of symmetric distributions, the asymptotic null distribution of the proposed class of statistics is shown to be chi-square. One particular member of this class of tests is recommended. This specific statistic is obtained using a linear function of the ranks of Euclidean norms of the transformed vectors. Comparisons between this new test statistic and several leading competitors are made through efficiency calculations and Monte Carlo studies. This new statistic has strong efficiencies over a wide Spectrum of distributions, ranging from very light-tailed distributions to very heavy-tailed ones. It performs as well as and in many cases better than its competitors. This statistic is very easy to compute for data in any practical dimension. This distinguishes it from some of the other affine-invariant signed rank tests in the literature.