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

Paindaveine, Davy

doi:10.1016/j.stamet.2004.08.001

Cited by 6 publications

(7 citation statements)

References 33 publications

(41 reference statements)

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“…The Gaussian competitors here are of the Hotelling, Fisher, or Lagrange multiplier forms. For all those tests, Chernoff-Savage result similar to (1.1) have been established (see also [26,27]). Hodges-Lehmann results also have been obtained, with bounds that, quite interestingly, depend on the dimension of the observation space: see [7].…”

Section: Introductionsupporting

confidence: 53%

On Hodges and Lehmann’s “6/π Result”

Hallin

Swan

Verdebout

2013

Contemporary Developments in Statistical Theory

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While the asymptotic relative efficiency (ARE) of Wilcoxon rank-based tests for location and regression with respect to their parametric Student competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that the ARE of the same Wilcoxon tests with respect to their van der Waerden or normal-score counterparts is bounded from above by 6/π ≈ 1.910. In this paper, we revisit that result, and investigate similar bounds for statistics based on Student scores. We also consider the serial version of this ARE. More precisely, we study the ARE, under various densities, of the Spearman-Wald-Wolfowitz and Kendall rank-based autocorrelations with respect to the van der Waerden or normal-score ones used to test (ARMA) serial dependence alternatives.

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Section: Introductionsupporting

confidence: 53%

On Hodges and Lehmann’s “6/π Result”

Hallin

Swan

Verdebout

2013

Contemporary Developments in Statistical Theory

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“…This sequence of lower bounds is monotonically decreasing for k ≥ 2, and goes to 0.648 as n → ∞ (see Hallin and Paindaveine (2002a) for the proofs of Corollaries 2 and 3, as well as for the densities in which the above infima are reached; see also Paindaveine (2004) for an elementary proof of Corollary 2).…”

Section: Proposition 3 Denote Byqmentioning

confidence: 94%

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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“…We then briefly explain why standard variational techniques are inappropriate for the problem under study, and eventually give a proof of Theorem 1 that is essentially based on Cauchy-Schwarz inequality, Jensen's inequality, and the arithmetic-geometric mean inequality (the latter-which, incidentally, is a particular case of Jensen's inequality for some appropriate convex function and discrete measure-plays, in the proof of Theorem 1, the same role as the arithmetic-harmonic mean inequality in the proof of Chernoff-Savage results for multivariate location; see [18]). …”

Section: Proof Of Theoremmentioning

confidence: 99%

“…See also Paindaveine [18] for a proof a la Gastwirth and Wolff [3] of multivariate Chernoff-Savage results for location parameters.…”

Section: Inappropriateness Of Standard Variational Argumentsmentioning

confidence: 99%

A Chernoff–Savage result for shape:On the non-admissibility of pseudo-Gaussian methods

Paindaveine

2006

Journal of Multivariate Analysis

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Chernoff and Savage [Asymptotic normality and efficiency of certain non-parametric tests, Ann. Math. Statist. 29 (1958) 972-994] established that, in the context of univariate location models, Gaussian-score rank-based procedures uniformly dominate-in terms of Pitman asymptotic relative efficiencies-their pseudo-Gaussian parametric counterparts. This result, which had quite an impact on the success and subsequent development of rank-based inference, has been extended to many location problems, including problems involving multivariate and/or dependent observations. In this paper, we show that this uniform dominance also holds in problems for which the parameter of interest is the shape of an elliptical distribution. The Pitman non-admissibility of the pseudo-Gaussian maximum likelihood estimator for shape and that of the pseudo-Gaussian likelihood ratio test of sphericity follow.

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

Cited by 6 publications

References 33 publications

On Hodges and Lehmann’s “6/π Result”

On Hodges and Lehmann’s “6/π Result”

Optimal signed-rank tests based on hyperplanes

A Chernoff–Savage result for shape:On the non-admissibility of pseudo-Gaussian methods

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