Optimal signed-rank tests based on hyperplanes

Abstract

For analysing k-variate data sets, Randles (J. Amer. Statist. Assoc. 84 (1989) 1045) considered hyperplanes going through k - 1 data points and the origin. He then introduced an empirical angular distance between two k-variate data vectors based on the number of hyperplanes (the so-called interdirections) that separate these two points, and proposed a multivariate sign test based on those interdirections. In this paper, we present an analogous concept (namely, lift-interdirections) to measure the regular distances between data points. The empirical distance between two k-variate data vectors is again determined by the number of hyperplanes that separate these two points; in this case, however, the considered hyperplanes are going through k distinct data points. The invariance and convergence properties of the empirical distances are considered. We show that the lift-interdirections together with Randles’ interdirections allow for building hyperplane-based versions of the optimal testing procedures developed in Hallin and Paindaveine (Ann. Statist. 30 (2002a) 1103, Statistical Data Analysis Based on the L 1 -Norm and Related Procedures (2002b) Birkhäuser, Basel, pp. 271; Bernoulli 8 (2002c) 787, Ann. Statist. (2004a) to appear) for a broad class of location and time series problems. The resulting procedures, which generalize the univariate signed-rank procedures, are affine-invariant and asymptotically invariant under a group of monotone radial transformations (acting on the standardized residuals). Consequently, they are asymptotically distribution-free under the class of elliptical distributions. They are optimal under correctly specified radial densities and, in several cases, enjoy a uniformly good efficiency behavior. These asymptotic properties are confirmed by a Monte-Carlo study, and, finally, a simple robustness study is conducted. It is remarkable that, in the test construction, the value of the test statistic depends on the data cloud only through the geometrical notions of data vectors and oriented hyperplanes, and their relations “above” and “below”.