Abstract
Based on the multivariate spatial rank function introduced by Möttönen and Oja [(1995), ‘Multivariate Spatial Sign and Rank Methods’, Journal of Nonparametric Statistics, 5, 201–213] and Möttönen et al. [(1997), ‘On the Efficiency of Multivariate Spatial Sign and Rank Tests’, Annals of Statistics, 25, 542–552], an extension of the univariate Wilcoxon regression estimate to multivariate linear models is proposed and studied. For both of the cases covariates are deterministic and i.i.d. random: we show that the proposed estimate is consistent and asymptotically normal under some appropriate assumptions. We have demonstrated that the asymptotic relative efficiency of the new regression estimate is the same as that of the generalised multivariate Hodges–Lehmann location estimates proposed by Chaudhuri [(1992), ‘Multivariate Location Estimation Using Extension of R-estimates Through U-statistics Type Approach’, Annals of Statistics, 20, 897–916] (with m=2); thus it possesses high efficiency. Simulations show that it also performs very well in the finite sample data. While the estimate is only rotation invariant, a version that is affine invariant is proposed as well.
Published Version
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