Statistica Sinica

Biman Chakraborty, Probal Chaudhuri and Hannu Oja^{*}

Abstract:An affine equivariant modification of the spatial median constructed using an adaptive transformation and retransformation procedure has been studied. It has been shown that this new estimate of multivariate location improves upon the performance of nonequivariant spatial median especially when there are correlations among the real valued components of multivariate data as well as when the scales of those components are different (e.g. when data points follow an elliptically symmetric distribution). For such correlated multivariate data the proposed estimate is more efficient than the non-equivariant vector of coordinatewise sample medians, and it outperforms the sample mean vector in the case of heavy tailed non-normal distributions. As an extension of the methodology, we have proposed an affine invariant modification of the well-known angle test based on the transformation approach, which is applicable to one sample multivariate location problems. We have observed that this affine invariant test performs better than the noninvariant angle test and the coordinatewise sign test for correlated multivariate data. Also, for heavy tailed non-normal multivariate distributions, the test outperforms Hotelling'sT^{2}test. Finite sample performance of the proposed estimate and the test is investigated using Monte Carlo simulations. Some data analytic examples are presented to demonstrate the implementation of the methodology in practice.

Key words and phrases:Affine transformation, efficiency, elliptically symmetric distribution, equivariant estimate, invariant test, multivariate median, multivariate sign test.