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Abstract:
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Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified $d$-dimensional error density has been an open problem for more than half a century. None of the solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently (Hallin, del Barrio, Cuesta-Albertos, and Matran, Ann. Statist., in press). Center-outward ranks and signs are not only distribution-free but achieve in dimension $d$ the (essential) maximal ancillarity property of traditional univariate ranks, hence carry all the "distribution-free information" available in the sample. We derive here the Hajek representation and asymptotic normality results required in the construction of center-outward rank tests for multiple-out regression and MANOVA. When based on appropriate spherical scores, these fully distribution-free tests achieve parametric efficiency.
This is joint work with D. Hlubinka and S. Hudecova, Charles University, Prague.
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