On the Efficiency of Affine Invariant Multivariate Rank Tests

Möttönen, Jyrki; Hettmansperger, Thomas P.; Oja, Hannu; Tienari, Juha

doi:10.1006/jmva.1998.1740

Cited by 17 publications

(13 citation statements)

References 24 publications

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“…v (z) ≤ 1 for all z ∈ R k , Lebesgue's dominated convergence theorem allows to conclude that (18) is o(1), as n → ∞, i.e., that (16) is o(1) as n → ∞. One can similarly prove that (17) is o(1) as n → ∞.…”

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confidence: 88%

Optimal signed-rank tests based on hyperplanes

Oja

Paindaveine

2005

Journal of Statistical Planning and Inference

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For analysing k-variate data sets, Randles (1989) 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 (2002a, b, c, and 2004a) 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". MSC: 62G10; 62G35

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confidence: 88%

Optimal signed-rank tests based on hyperplanes

Oja

Paindaveine

2005

Journal of Statistical Planning and Inference

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“…The same optimality property still holds for (S3)-but not for (S2), which is only rotationally invariant-under elliptical symmetry (still, with double-exponential radial density). The reader is referred to Möttönen et al (1997Möttönen et al ( , 1998 for the derivation of ARE values for (SR) and (OR).…”

Section: Asymptotic Relative Efficienciesmentioning

confidence: 99%

“…The first of these groups relies on componentwise rankings [see, e.g., the monograph by Puri and Sen (1971)], but suffers from a severe lack of invariance with respect to affine transformations, which has been the main motivation for the other two approaches. The second group [Möttönen et al (1995[Möttönen et al ( , 1998; Hettmansperger et al (1994Hettmansperger et al ( , 1997; see Oja (1999) for a recent review] is closely related with the spatial signs and ranks, and with the so-called Oja median, the third one [Randles (1989); Peters and Randles (1990); Jan and Randles (1994)] with the concept of interdirections.…”

Section: Introduction and Main Assumptions Denote By (X (N)mentioning

confidence: 99%

Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks

Hallin¹,

Paindaveine²

2002

Ann. Statist.

100

121

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We propose a family of tests, based on Randles' (1989) concept of interdirections and the ranks of pseudo-Mahalanobis distances computed with respect to a multivariate M-estimator of scatter due to Tyler (1987), for the multivariate one-sample problem under elliptical symmetry. These tests, which generalize the univariate signed-rank tests, are affine-invariant. Depending on the score function considered (van der Waerden, Laplace, . . .), they allow for locally asymptotically maximin tests at selected densities (multivariate normal, multivariate double-exponential, . . .). Local powers and asymptotic relative efficiencies are derived-with respect to Hotelling's test, Randles' (1989) multivariate sign test, Peters and Randles' (1990) Wilcoxon-type test, and with respect to the Oja median tests. We, moreover, extend to the multivariate setting two famous univariate results: the traditional Chernoff-Savage (1958) property, showing that Hotelling's traditional procedure is uniformly dominated, in the Pitman sense, by the van der Waerden version of our tests, and the celebrated Hodges-Lehmann (1956) ".864 result," providing, for any fixed space dimension k, the lower bound for the asymptotic relative efficiency of Wilcoxon-type tests with respect to Hotelling's.These asymptotic results are confirmed by a Monte Carlo investigation, and application to a real data set.

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“…The first one, based on Oja signs and ranks, is due to Oja, Hettmansperger, and their collaborators (Möttönen et al [18][19][20]; Hettmansperger et al [21,22]; see Oja [23] for a review). The second one is associated with ranks of Mahalanobis distances and Randles' concept of interdirections, and was developed by Randles and his coauthors (Randles [24,25]; Peters and Randles [26]; Jan and Randles [27]; Randles and Um [28]).…”

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confidence: 99%

A unified and elementary proof of serial and nonserial, univariate and multivariate, Chernoff–Savage results

Paindaveine

2004

Statistical Methodology

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We provide a simple proof that the Chernoff-Savage [1] result, establishing the uniform dominance of normal-score rank procedures over their Gaussian competitors, also holds in a broad class of problems involving serial and/or multivariate observations. The non-admissibility of the corresponding everyday practice Gaussian procedures (multivariate least-squares estimators, multivariate t-tests and F -tests, correlogram-based methods, multivariate portmanteau and Durbin-Watson tests, etc.) follows. The proof, which generalizes to the multivariate-possibly serial-setup the idea developed in Gastwirth and Wolff [2] in the context of univariate location problems, allows for avoiding technical convexity and variational arguments.

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On the Efficiency of Affine Invariant Multivariate Rank Tests

Cited by 17 publications

References 24 publications

Optimal signed-rank tests based on hyperplanes

Optimal signed-rank tests based on hyperplanes

Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks

A unified and elementary proof of serial and nonserial, univariate and multivariate, Chernoff–Savage results

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