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) Birkhauser, 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”.

[1]  Hannu Oja,et al.  The geometry of the affine invariant multivariate sign and rank methods , 1999 .

[2]  D. Paindaveine,et al.  Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks , 2002 .

[3]  Herman Chernoff,et al.  ASYMPTOTIC NORMALITY AND EFFICIENCY OF CERTAIN NONPARAMETRIC TEST STATISTICS , 1958 .

[4]  Ronald H. Randles,et al.  A Simpler, Affine-Invariant, Multivariate, Distribution-Free Sign Test , 2000 .

[5]  Marc Hallin,et al.  Rank-based optimal tests of the adequacy of an elliptic VARMA model , 2004 .

[6]  D. Paindaveine.  A unified and elementary proof of serial and nonserial, univariate and multivariate: Chernoff-Savage results , 2004 .

[7]  Hannu Oja,et al.  On the Efficiency of Affine Invariant Multivariate Rank Tests , 1998 .

[8]  Hannu Oja,et al.  Affine Invariant Multivariate One‐Sample Sign Tests , 1994 .

[9]  David E. Tyler.  A Distribution-Free $M$-Estimator of Multivariate Scatter , 1987 .

[10]  D. Paindaveine,et al.  Multivariate signed ranks: Randles' interdirections or Tyler's angles? , 2002 .

[11]  Marc Hallin,et al.  Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors , 2005 .

[12]  M. Bilodeau,et al.  Theory of multivariate statistics , 1999 .

[13]  Ronald H. Randles,et al.  A Multivariate Signed-Rank Test for the One-Sample Location Problem , 1990 .

[14]  H. Oja,et al.  A scatter matrix estimate based on the zonotope , 2003 .

[15]  Hannu Oja,et al.  Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review , 1999 .

[16]  Hannu Oja,et al.  Affine-invariant multivariate one-sample signed-rank tests , 1997 .

[17]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[18]  Ronald H. Randles,et al.  NONPARAMETRIC TESTS FOR THE MULTIVARIATE MULTI-SAMPLE LOCATION PROBLEM , 2003 .

[19]  Hannu Oja,et al.  ON THE EFFICIENCY OF MULTIVARIATE SPATIAL SIGN AND RANK TESTS , 1997 .

[20]  R. Randles.  A Distribution-Free Multivariate Sign Test Based on Interdirections , 1989 .

[21]  D. Paindaveine,et al.  Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence , 2002 .

[22]  P. Sen,et al.  Nonparametric methods in multivariate analysis , 1974 .

[23]  J. L. Hodges,et al.  The Efficiency of Some Nonparametric Competitors of the t-Test , 1956 .

[24]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .