On multivariate median regression

An extension of the concept of least absolute deviation regression for prob lems with multivariate response is considered. The approach is based on a t ransformation and retransformation technique that chooses a data-driven coo rdinate system for transforming the response vectors-and then retransforms the estimate of the matrix of regression parameters, which is obtained by p erforming coordinatewise least absolute deviations regression on the transf ormed response vectors. It is shown that the estimates are equivariant unde r non-singular linear transformations of the response vectors. An algorithm called TREMMER (Transformation Retransformation Estimates in Multivariate MEdian:Regression) has been suggested which adaptively chooses the optimal data-driven coordinate system and then computes the regression estimates. W e have also indicated how resampling techniques like the bootstrap can be u sed to conveniently estimate the standard errors of TREMMER estimates. It i s shown that the proposed estimate is more efficient than the non-equivaria nt coordinatewise least absolute deviations estimate, and it out performs o rdinary least-squares estimates in the case of heavy-tailed non-normal mult ivariate error distributions, Asymptotic normality and some other optimalit y properties of the estimate are also discussed. Some interesting examples are presented to motivate the need for affine equivariant estimation in mul tivariate median regression and to demonstrate,the performance of the propo sed methodology.