On Certain Bivariate Sign Tests and Medians

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Abstract
Brown and Hettmansperger proposed an affine invariant bivariate analogue of the sign test, the OS test, based on the generalized median of Oja. On the other hand, Oja and Nyblom introduced a family of locally most powerful affine invariant sign tests. In the case of elliptic distributions, the locally most powerful Blumen's test and the OS test are shown to be asymptotically equivalent. Formulas for calculating asymptotic relative efficiencies of the OS test and the Oja bivariate median are given. It is shown that if the contours of a distribution are of a similar shape, the relative efficiencies of the OS test and Blumen's test depend on the distribution only through the shape of the contours. For the power family of contours |x 1| p + |x 2| p = c, p > 0, numerical calculations show that the efficiency of the OS test relative to Blumen's test attains its minimum 1 as p = 2 (spherical/elliptic case) and increases to infinity as p → 0. In the bivariate Laplace case with independent marginals (p = 1) as well as in the case p = ∞ the relative efficiency is 1.028.

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