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## Semiparametrically efficient rank-based inference for shape I: Optimal rank-based tests for sphericity (2006)

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Venue: | Ann. Statist |

Citations: | 48 - 32 self |

### Citations

2198 |
An introduction to multivariate statistical analysis, second edition
- Anderson
- 1984
(Show Context)
Citation Context ...Xi) with location center θ = (θ1,...,θk) ′, scale σ2 , shape matrix V, and radial density g1 is defined to be 3κk(g1) := E[(Xi − θi) 4 ] E2 [(Xi − θi) 2 − 3; ] 1k (U))4 ] < ∞, where see, for example, =-=[1]-=-, page 54, [38] or [50]. This quantity depends only on the dimension k and the radial density g1, not on i or on the other parameters characterizing the elliptical distribution (which of course justif... |

719 |
Théorie des distributions
- Schwartz
- 1966
(Show Context)
Citation Context ...exp (x) − h(Df1/2 1;exp )(x)]2e kx dx = o(h 2 ) as h → 0. In that case, Df 1/2 1;exp and (f1/2 1;exp )′ are equal in L2 (R,νk). The proof of this lemma relies on the following result by Schwartz (see =-=[47]-=-, pages 186–188): Lemma A.3 (Schwartz). The real function g is in W 1,2 (R) (with weak derivative g ′ , say) iff (i) g ∈ L 2 (R) and (ii) there exists Dg ∈ L 2 (R) such that x ↦→ g(x + h) − g(x) − h(D... |

550 | Matrix Differential Calculus with Applications in Statistics and Econometrics - Magnus, Neudecker - 1999 |

434 |
Statistics of Directional Data.
- Mardia
- 1972
(Show Context)
Citation Context ...) for (X − θ)—are also of direct interest in several specific domains of application such as geostatistics, paleomagnetic studies in geology, animal navigation, astronomy and wind direction data; see =-=[5, 34, 53]-=- or [35] for references. Finally, shape matrices provide robust alternatives to traditional covariance matrices; as such, they are obvious candidates for serving as the basic tools in a host of multiv... |

426 |
Lucien Asymptotic methods in statistical decision theory. Springer Series in Statistics. New York etc.:
- Cam
- 1986
(Show Context)
Citation Context ...oot-n consistent preliminary estimators into uniformly root-n consistent ones (see, e.g., Lemma 4.4 in [30] for a typical use), are quite standard in Le Cam’s one-step construction of estimators (see =-=[31]-=-), and several of them, characterized by a # subscript, will appear in the sequel. Denoting by ⌈x⌉ the smallest integer larger than or equal to x and by c0 an arbitrary positive constant that does not... |

418 |
Directional Statistics.
- Mardia, Jupp
- 2000
(Show Context)
Citation Context ...re also of direct interest in several specific domains of application such as geostatistics, paleomagnetic studies in geology, animal navigation, astronomy and wind direction data; see [5, 34, 53] or =-=[35]-=- for references. Finally, shape matrices provide robust alternatives to traditional covariance matrices; as such, they are obvious candidates for serving as the basic tools in a host of multivariate a... |

375 |
A class of statistics with asymptotically normal distribution
- Hoeffding
- 1948
(Show Context)
Citation Context ... uu ′ − 1 k Ik )+ ℓ ( (u,v) ↦→ E[I[d2 ≤ d1]|U1 = u,U2 = v]vec uu ′ − 1 k Ik ) − , ℓ respectively. The continuous mapping theorem and standard asymptotic normality results for U-statistics (see, e.g., =-=[21]-=-) imply that E (n) 2;ℓ ≤ n1/2 [ ( E vec U1U ′ 1 − 1 k Ik ) −] (A.3) ℓ { E[K(P[d2 ≤ d1|d1,U1,U2])vec(U1U × ′ + 1 − (1/k)Ik) ℓ ] E[vec(U1U ′ 1 − (1/k)Ik) − ℓ ] ( E[I[d2 ≤ d1]vec(U1U − K ′ − 1 − (1/k)Ik)... |

150 | Statistical applications of the multivariate skew normal distribution.
- Azzalini, Capatanio
- 1999
(Show Context)
Citation Context ...′ v) −1/2 mv, with v = (0.15,0) ′ . The distribution of the resulting Xi’s is the so-called bivariate skew normal distribution with parameters 0, I2 and mv (see, e.g.,36 M. HALLIN AND D. PAINDAVEINE =-=[3]-=- or [4]). Population St2 is obtained in the same way, but with trivariate t2distributed vectors (Vm;i,W ′ m;i )′ with the same mean and covariance matrix as in the Gaussian case above, but v = (0.25,0... |

143 | Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution.
- Azzalini, Capitanio
- 2003
(Show Context)
Citation Context .../2 mv, with v = (0.15,0) ′ . The distribution of the resulting Xi’s is the so-called bivariate skew normal distribution with parameters 0, I2 and mv (see, e.g.,36 M. HALLIN AND D. PAINDAVEINE [3] or =-=[4]-=-). Population St2 is obtained in the same way, but with trivariate t2distributed vectors (Vm;i,W ′ m;i )′ with the same mean and covariance matrix as in the Gaussian case above, but v = (0.25,0) ′ (se... |

121 |
On adaptive estimation.
- Bickel
- 1982
(Show Context)
Citation Context ... tool is the uniform local asymptotic normality (ULAN), with respect to ϑ = (θ ′ ,σ2 ◦ ,( vechV) ′)′, of the families P (n) . This LAN (ULAN) issue has been briefly touched by f1 Bickel (Example 4 in =-=[7]-=-). The very particular case of bivariate distributions with finite second-order moments has been treated recently by Falk [11] in his investigation of the inefficiency of empirical correlation coeffic... |

110 |
A distribution-free M-estimator of multivariate scatter,”
- Tyler
- 1987
(Show Context)
Citation Context ...ssian estimators, including (iv) the Chernoff–Savage property of [38]? Such estimators would improve the performance of the existing ones that satisfy the consistency requirement (i), such as Tyler’s =-=[45]-=- celebrated affine-equivariant estimator of shape (scatter, in Tyler’s terminology) V (n) T or the estimator of shape based on the Oja signs developed in [36]. These estimators are indeed root-n consi... |

80 |
On a geometric notion of quantiles for multivariate data,
- Chaudhuri
- 1996
(Show Context)
Citation Context ...multivariate sign- and rank-based competitors of the Gaussian likelihood procedures are proposed. The ranks used by the authors are the spatial ranks introduced by Möttönen and Oja [37] and Chaudhuri =-=[9]-=-; see also [32]. Although Pitman efficiencies (with respect to the Gaussian methods) are obtained, no attempt is made to achieve any optimality and the authors restrict themselves to procedures of the... |

79 | Theory of Multivariate Statistics,
- Bilodeau, Brenner
- 1999
(Show Context)
Citation Context ...id from the point of view of type I risk, such a test, for the null hypothesis of i.i.d.ness, would be severely biased and inconsistent. Indeed the Maxwell–Hershell theorem (see, e.g., pages 51–52 of =-=[8]-=-) indicates that all non-Gaussian spherical distributions are part of the alternative, while our Proposition 5.1(v) establishes that α-level extended sign tests at spherical alternatives have asymptot... |

77 |
Significance test for sphericity of a normal n-variate distribution.
- Mauchly
- 1940
(Show Context)
Citation Context ...on-Gaussian g1’s and, quite remarkably, (iv) whenEFFICIENT RANK-BASED INFERENCE FOR SHAPE II 3 Gaussian (van der Waerden) scores are adopted, their ARE’s with respect to the classical Gaussian tests =-=[21, 22, 34, 35]-=- are uniformly larger than one; see [38] for this extension of the celebrated Chernoff–Savage [5] result to shape matrices. These optimality properties, in fact, are all possessed by the noncentrality... |

76 |
Estimating integrated squared density derivatives: sharp best order of convergence estimates.
- Bickel, Ritov
- 1988
(Show Context)
Citation Context ...es involve a kernel estimate of g1 and, hence, cannot be expected to perform well under small and moderate sample sizes. Such kernel methods have been considered for Wilcoxon scores in [41] (see also =-=[3, 4, 7]-=- and, in a more general setting, in Section 4.5 of [27]. They also require arbitrary choices (window width and kernel or, as in [27], the choice of the order α of an empirical quantile) for which univ... |

72 |
On adaptive estimation in stationary ARMA processes,
- Kreiss
- 1987
(Show Context)
Citation Context ...hape k Ik. Denote by V (n) # a discretized version of V(n) T . Such discretizations, which turn root-n consistent preliminary estimators into uniformly root-n consistent ones (see, e.g., Lemma 4.4 in =-=[30]-=- for a typical use), are quite standard in Le Cam’s one-step construction of estimators (see [31]), and several of them, characterized by a # subscript, will appear in the sequel. Denoting by ⌈x⌉ the ... |

68 |
Asymptotic Normality and Efficiency of Certain Nonparametric Tests
- Chernoff, Savage
- 1958
(Show Context)
Citation Context ...der Waerden) scores are adopted, their ARE’s with respect to the classical Gaussian tests [21, 22, 34, 35] are uniformly larger than one; see [38] for this extension of the celebrated Chernoff–Savage =-=[5]-=- result to shape matrices. These optimality properties, in fact, are all possessed by the noncentrality parameters of the noncentral chi-square asymptotic distributions, under local alternatives, of t... |

60 |
Estimating regression coefficients by minimizing the dispersion of the residuals,
- Jaeckel
- 1972
(Show Context)
Citation Context ...tion. The derivation of such R-estimators, however, is by no means straightforward. Traditional R-estimators are defined (and computed) via the minimization of some rank-based objective function; see =-=[1, 19, 20, 24, 26]-=- or the review paper by Draper [6]. In the present context, this approach, in connection with (1.1), leads to the definition of an R-estimator as (1.2) (n) V := argmin Q ∼f1 f1 V ∼ ( (V) = argmin tr(S... |

50 |
Estimates of location based on rank tests’,
- Hodges, Lehmann
- 1963
(Show Context)
Citation Context ...tion. The derivation of such R-estimators, however, is by no means straightforward. Traditional R-estimators are defined (and computed) via the minimization of some rank-based objective function; see =-=[1, 19, 20, 24, 26]-=- or the review paper by Draper [6]. In the present context, this approach, in connection with (1.1), leads to the definition of an R-estimator as (1.2) (n) V := argmin Q ∼f1 f1 V ∼ ( (V) = argmin tr(S... |

47 |
Nonparametric methods in general linear models,
- Puri, Sen
- 1985
(Show Context)
Citation Context ...= 2(k − 1)/(k2 (k + 2)) for ℓ ∈ Lk := {mk + m + 1,m = 0,1,...,k − 1} and Cℓ,k = Var[U1,1U1,2] = 1/k2 for ℓ /∈ Lk. Hájek’s classical projection result for linear signed rank statistics ([15]; see also =-=[44]-=-, Chapter 3) thus yields the desired result. □ Proof of Proposition 4.1. From Lemma 4.1, we easily obtain [for ◦ vechV0) ′) of the parameter] any fixed value ϑ ′ 0 := (θ ′ ,σ 2 ,( (n) Q K = (∆⋆(n) K;g... |

46 |
Asymptotic normality of simple linear rank statistics under alternatives
- Hájek
- 1968
(Show Context)
Citation Context ...conditional on U (n) , each component T ˜ (n) K,ℓ of T (n) K is a linear rank ˜ statistic with approximate scores K( i n+1 ). Under the assumptions made, the Hájek variance inequality (Theorem 3.1 in =-=[14]-=-) applies (conditional on U (n)), yielding, for all ℓ [with appropriate r and s, m (n) 1 ∑ni=1 K := n K( i n+1 ) and σ 2 K := ∫ 1 0 K2 (u)du − ( ∫ 1 Var(T ˜ (A.2) (n) K,ℓ |U(n) ) ≤ 21 max 1≤i≤n < 21σ ... |

39 |
Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations”,
- Muirhead, Waternaux
- 1980
(Show Context)
Citation Context ...on-Gaussian g1’s and, quite remarkably, (iv) whenEFFICIENT RANK-BASED INFERENCE FOR SHAPE II 3 Gaussian (van der Waerden) scores are adopted, their ARE’s with respect to the classical Gaussian tests =-=[21, 22, 34, 35]-=- are uniformly larger than one; see [38] for this extension of the celebrated Chernoff–Savage [5] result to shape matrices. These optimality properties, in fact, are all possessed by the noncentrality... |

39 |
On the asymptotic normality of statistics with estimated parameters.
- Randles
- 1982
(Show Context)
Citation Context ...Möttönen and Oja [37]), which is itself “sign-based.” The asymptotic impact of this substitution on the validity of the signed rank tests proposed in Section 4.2 could be studied directly (see, e.g., =-=[45]-=-),OPTIMAL RANK-BASED TESTS FOR SPHERICITY 29 Table 2 AREs of the t6-, van der Waerden-, sign- and Wilcoxon-score rank-based tests for shape and (in parentheses) location, with respect to the correspo... |

39 |
Robustness and efficiency properties of scatter matrices”,
- Tyler
- 1983
(Show Context)
Citation Context .../2 [ Ik2 +Kk − 2 k Jk ] (V ⊗2 ) −1/2 M ′ k =: Jk(f1)Υ −1 k (V), a form that is not unfamiliar in the area of robust estimation of covariance matrices; see, for instance, the asymptotic covariances in =-=[40, 42, 50, 51]-=- for the covariances of scatter estimates [as in (2.6), (2.7)], [41, 52] for covariances of shape estimates [as in (3.2)]. In the sequel, optimality (in the local and asymptotic sense, at radial densi... |

38 | On the estimation of quadratic functionals.
- Fan
- 1991
(Show Context)
Citation Context ... involve a kernel estimate of g1 and, hence, cannot be expected to perform well under small and moderate sample sizes. Such kernel methods have been considered, for Wilcoxon scores, by [41] (see also =-=[3, 4, 7]-=- and, in a more general setting, in Section 4.5 of [27]. They also require arbitrary choices (window width and kernel or, as in [27], the choice of the order α of an empirical quantile) for which univ... |

38 |
Multivariate spatial sign and rank methods.
- Mottonen, Oja
- 1995
(Show Context)
Citation Context ...shape. Appropriate multivariate sign- and rank-based competitors of the Gaussian likelihood procedures are proposed. The ranks used by the authors are the spatial ranks introduced by Möttönen and Oja =-=[37]-=- and Chaudhuri [9]; see also [32]. Although Pitman efficiencies (with respect to the Gaussian methods) are obtained, no attempt is made to achieve any optimality and the authors restrict themselves to... |

36 |
Nonparametric estimate of regression coefficients
- Jurečková
- 1971
(Show Context)
Citation Context ...tion. The derivation of such R-estimators, however, is by no means straightforward. Traditional R-estimators are defined (and computed) via the minimization of some rank-based objective function; see =-=[1, 19, 20, 24, 26]-=- or the review paper by Draper [6]. In the present context, this approach, in connection with (1.1), leads to the definition of an R-estimator as (1.2) (n) V := argmin Q ∼f1 f1 V ∼ ( (V) = argmin tr(S... |

34 |
Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis
- Hallin, Paindaveine
- 2002
(Show Context)
Citation Context ...restrictions. In such a context, robust inference methods, resisting arbitrarily heavy radial tails, are highly desirable and distribution-free rank-based methods naturally come into the picture (see =-=[9, 10, 11, 12]-=- for closely related results). 1.2. Rank tests. In the hypothesis-testing context, HP develop a class of semiparametrically optimal signed rank tests for null hypotheses of the form V = V0 (θ, σ and g... |

34 |
A practical affine equivariant multivariate median.
- Hettmansperger, Randles
- 2003
(Show Context)
Citation Context ...(n) σ2 ⋃ := ;f1 f1∈FA σ>0 σ 2 ,V;f1 Although any root-n consistent estimator ˆ θ could be used, we suggest adopting the multivariate affine-equivariant median introduced by Hettmansperger and Randles =-=[18]-=- which is itself a “sign-based” estimator. The multivariate signs to be considered, then, are the U (n) i ( ˆ θ,V)’s and the ranks to be considered are those of the d (n) i ( ˆ θ,V)’s. 2.2. Semiparame... |

33 |
Asymptotic linearity of a rank statistic in regression parameter,
- Jureckova
- 1969
(Show Context)
Citation Context ...i) The claim in part (ii) of the lemma is the same as in part (i), except that dn i and Uni replace d0i and U0i , respectively. Accordingly, part (ii) holds under P (n) θ n ,σ 2 ,V n ;g1 Lemma 3.5 of =-=[23]-=-. (iii) Note that |Jk(f1,g1)−J (ℓ) . That it also holds under P(n) θ,σ 2 ,V;g1 follows from (Kf1(u)−K(ℓ) (u))Kg1(u)du|2 k (f1;g1)| 2 = | ∫ 1 0 f1 ≤ Jk(g1) ∫ 1 |Kf1 0 (u) − K(ℓ) f1 (u)|2 du. Again, |K ... |

33 |
Asymptotics of Reweighted Estimators of Multivariate Location
- Lopuhaa
- 1999
(Show Context)
Citation Context ...g, as the asymptotic covariance matrices of (the vec versions of) ̂ V ∼ Qk(V). Their relative performances can thus be described by a single number, a fact that was already observed in [44] (see also =-=[33]-=-); the situation is entirely different for covariance matrices, where two numbers are required [36, 37, 43]. These ARE’s coincide with those obtained in HP for the problem of testing V = V0 (see Propo... |

31 |
Non-parametric confidence intervals for a shift parameter.
- Lehmann
- 1963
(Show Context)
Citation Context ...een as popular as rank tests in applications. Simple consistent estimators of cross-information coefficients (the definition of which depends on the problem under study) have been proposed by Lehmann =-=[32]-=- and Sen [42] for one- and two-sample location problems; these estimators are based on comparisons of confidence interval lengths, a method involving the arbitrary choice of a confidence level (1 −α) ... |

29 |
Affine invariant multivariate sign and rank tests and corresponding estimates: A review.
- Oja
- 1999
(Show Context)
Citation Context ...s on statistics that are measurable with respect to invariant or distribution-free quantities such as the multivariate concepts of signs and ranks developed, mainly, in the robustness literature; see =-=[39]-=- for a review. This sign-and/or-rank-based approach has been adopted by Tyler [53], Ghosh and Sengupta [13] and Marden and Gao [33]. Tyler [53] addresses the problem of testing uniformity over the sph... |

28 |
Some optimal multivariate tests.
- John
- 1971
(Show Context)
Citation Context ...on-Gaussian g1’s and, quite remarkably, (iv) whenEFFICIENT RANK-BASED INFERENCE FOR SHAPE II 3 Gaussian (van der Waerden) scores are adopted, their ARE’s with respect to the classical Gaussian tests =-=[21, 22, 34, 35]-=- are uniformly larger than one; see [38] for this extension of the celebrated Chernoff–Savage [5] result to shape matrices. These optimality properties, in fact, are all possessed by the noncentrality... |

28 |
The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend,
- Swensen
- 1985
(Show Context)
Citation Context ...on seems to produce strictly conservative critical values for the van der Waerden- and t6-score versions of our tests. APPENDIX A.1. Proof of Proposition 2.1. Our proof relies on Lemma 1 from Swensen =-=[49]-=- (more precisely, on its extension by Garel and Hallin [12]). The sufficient conditions for LAN given in Swensen’s result follow from standard (x) is differentiable is the density in (1.1), and we the... |

27 |
A simpler, affine-invariant, multivariate, distribution-free sign test
- Randles
- 2000
(Show Context)
Citation Context ...If V(n) is strictly affine-equivariant [in the sense of (5.1)], then using the same notation as in Section 5, (V(n)(M,a))1/2 = dM(V(n))1/2O for some d > 0 and some k×k orthogonal matrix O (see, e.g., =-=[40]-=-). The strict affine-equivariance of the practical implementation V∼ (n) f1 = limm→∞ V∼ (n) f1#(c0,m) [which is based on V(n)T and W∼ (n) f1 (V(n)T ) instead of V (n) # and W∼ (n) f1#] follows. REFE... |

27 |
A distribution-free multivariate sign test based on interdirections
- Randles
- 1989
(Show Context)
Citation Context ...nd are simple to compute. The sign test statistic Q ˜ S essentially coincides with that proposed by Ghosh and Sengupta [13] where, however, the U ′ iUj are compared from Randles’ interdirections (see =-=[46]-=-). Local asymptotic optimality under radial density f1 is achieved by the test based on Q ˜ (4.7) k(k + 2) Q f1 = ˜ 2nJk(f1) which, letting Sf1 f1 := Q ˜ Kf 1 . This test statistic takes the form n∑ i... |

27 |
Statistical analysis for the angular central Gaussian distribution on the sphere
- TYLER
- 1987
(Show Context)
Citation Context ...) for (X − θ)—are also of direct interest in several specific domains of application such as geostatistics, paleomagnetic studies in geology, animal navigation, astronomy and wind direction data; see =-=[5, 34, 53]-=- or [35] for references. Finally, shape matrices provide robust alternatives to traditional covariance matrices; as such, they are obvious candidates for serving as the basic tools in a host of multiv... |

26 |
Semi-parametric efficiency, distribution-freeness and invariance
- Hallin, Werker
- 2003
(Show Context)
Citation Context ...t is, on the arbitrary choice of the normalization V11 = 1), the semiparametric information bound does not; see Sections HP3.1, HP3.2, [14] and [39] for details. A general result by Hallin and Werker =-=[17]-=- suggests that, in case (i) for all f1 ∈ FA and σ > 0, the sequence of parametric subexperiments P (n) σ2 [see (2.2)] is ULAN with central sequence ∆(n) ;f1 f1 (σ2,V) and information matrix Γf1 (σ2,V)... |

26 |
Local asymptotic normality of multivariate ARMA processes with a linear trend,
- Garel, Hallin
- 1995
(Show Context)
Citation Context ...or the van der Waerden- and t6-score versions of our tests. APPENDIX A.1. Proof of Proposition 2.1. Our proof relies on Lemma 1 from Swensen [49] (more precisely, on its extension by Garel and Hallin =-=[12]-=-). The sufficient conditions for LAN given in Swensen’s result follow from standard (x) is differentiable is the density in (1.1), and we therefore focus on this. The main step in establishing this qu... |

21 | Rank-based optimal tests of the adequacy of an elliptic VARMA model, The Annals of Statistics 32
- Hallin, Paindaveine
- 2004
(Show Context)
Citation Context ...restrictions. In such a context, robust inference methods, resisting arbitrarily heavy radial tails, are highly desirable and distribution-free rank-based methods naturally come into the picture (see =-=[9, 10, 11, 12]-=- for closely related results). 1.2. Rank tests. In the hypothesis-testing context, HP develop a class of semiparametrically optimal signed rank tests for null hypotheses of the form V = V0 (θ, σ and g... |

18 |
The distribution of a statistic used for testing sphericity of normal distributions.
- John
- 1972
(Show Context)
Citation Context |

18 |
Radial estimates and the test for sphericity
- Tyler
- 1982
(Show Context)
Citation Context ...rmances can thus be described by a single number, a fact that was already observed in [44] (see also [33]); the situation is entirely different for covariance matrices, where two numbers are required =-=[36, 37, 43]-=-. These ARE’s coincide with those obtained in HP for the problem of testing V = V0 (see Proposition HP4.2). An immediate corollary is that the Chernoff–Savage result of [38] also applies here: the ARE... |

18 |
Robust tests for spherical symmetry.
- Kariya, Eaton
- 1977
(Show Context)
Citation Context ...ed from the test statistics introduced by Tyler [50], who proposes replacing covariance matrices with more robust estimators of scatter. Non-Gaussian models have been investigated by Kariya and Eaton =-=[26]-=-, where elliptical densities, possibly with infinite variances, are considered. Uniformly most powerful unbiased tests are derived, basically against specified nonspherical shape values. The results o... |

15 |
On a distribution-free method of estimating asymptotic efficiency of a class of nonparametric tests
- Sen
- 1968
(Show Context)
Citation Context ...r as rank tests in applications. Simple consistent estimators of cross-information coefficients (the definition of which depends on the problem under study) have been proposed by Lehmann [32] and Sen =-=[42]-=- for one- and two-sample location problems; these estimators are based on comparisons of confidence interval lengths, a method involving the arbitrary choice of a confidence level (1 −α) which has qui... |

14 |
1967]. Estimates of regression parameters based on rank tests
- Adichie
(Show Context)
Citation Context |

13 | Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence
- Hallin, Paindaveine
- 2002
(Show Context)
Citation Context ...restrictions. In such a context, robust inference methods, resisting arbitrarily heavy radial tails, are highly desirable and distribution-free rank-based methods naturally come into the picture (see =-=[9, 10, 11, 12]-=- for closely related results). 1.2. Rank tests. In the hypothesis-testing context, HP develop a class of semiparametrically optimal signed rank tests for null hypotheses of the form V = V0 (θ, σ and g... |

13 |
Theory of rank tests. 2nd ed.
- Hajek, Sidak, et al.
- 1999
(Show Context)
Citation Context ... Var[U 2 1,1 ] = 2(k − 1)/(k2 (k + 2)) for ℓ ∈ Lk := {mk + m + 1,m = 0,1,...,k − 1} and Cℓ,k = Var[U1,1U1,2] = 1/k2 for ℓ /∈ Lk. Hájek’s classical projection result for linear signed rank statistics (=-=[15]-=-; see also [44], Chapter 3) thus yields the desired result. □ Proof of Proposition 4.1. From Lemma 4.1, we easily obtain [for ◦ vechV0) ′) of the parameter] any fixed value ϑ ′ 0 := (θ ′ ,σ 2 ,( (n) Q... |

12 |
Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors
- Hallin, Paindaveine
- 2005
(Show Context)
Citation Context |

12 |
Affine equivariant multivariate sign methods
- Ollila, Hettmansperger, et al.
- 2005
(Show Context)
Citation Context ...ot unfamiliar in the area of robust estimation of covariance matrices; see, for instance, the asymptotic covariances in [40, 42, 50, 51] for the covariances of scatter estimates [as in (2.6), (2.7)], =-=[41, 52]-=- for covariances of shape estimates [as in (3.2)]. In the sequel, optimality (in the local and asymptotic sense, at radial density f1) is to be understood in the context of the Gaussian shift experime... |

12 | A canonical definition of shape.
- Paindaveine
- 2008
(Show Context)
Citation Context ...whereas this loss depends on the definition of shape (that is, on the arbitrary choice of the normalization V11 = 1), the semiparametric information bound does not; see Sections HP3.1, HP3.2 [14] and =-=[39]-=- for details. A general result by Hallin and Werker [17] suggests that, in case (i) for all f1 ∈FA and σ > 0, the sequence of parametric subexperiments P(n)σ 2;f1 [see (2.2)] is ULAN with central sequ... |

11 |
Window estimation of the asymptotic variance of rank estimators of location.
- Schweder
- 1975
(Show Context)
Citation Context ...rate approaches involve a kernel estimate of g1 and, hence, cannot be expected to perform well under small and moderate sample sizes. Such kernel methods have been considered, for Wilcoxon scores, by =-=[41]-=- (see also [3, 4, 7] and, in a more general setting, in Section 4.5 of [27]. They also require arbitrary choices (window width and kernel or, as in [27], the choice of the order α of an empirical quan... |

10 | A Chernoff-Savage result for shape. On the nonadmissibility of pseudo-Gaussian methods
- Paindaveine
- 2006
(Show Context)
Citation Context ...FICIENT RANK-BASED INFERENCE FOR SHAPE II 3 Gaussian (van der Waerden) scores are adopted, their ARE’s with respect to the classical Gaussian tests [21, 22, 34, 35] are uniformly larger than one; see =-=[38]-=- for this extension of the celebrated Chernoff–Savage [5] result to shape matrices. These optimality properties, in fact, are all possessed by the noncentrality parameters of the noncentral chi-square... |

10 |
Testing for spherical symmetry of a multivariate distribution.
- Baringhaus
- 1991
(Show Context)
Citation Context ...) for (X − θ)—are also of direct interest in several specific domains of application such as geostatistics, paleomagnetic studies in geology, animal navigation, astronomy and wind direction data; see =-=[5, 34, 53]-=- or [35] for references. Finally, shape matrices provide robust alternatives to traditional covariance matrices; as such, they are obvious candidates for serving as the basic tools in a host of multiv... |

10 |
Locally best invariant test for sphericity and the limiting distributions
- Sugiura
- 1972
(Show Context)
Citation Context ...to the classical toolkit of multivariate analysis. The (Gaussian) locally most powerful invariant (under shift, scale and orthogonal transformations) test was obtained by John [24, 25] and by Sugiura =-=[48]-=-. In their original versions, these tests are valid under Gaussian assumptions only; however, with slight modifications, they remain valid under elliptical populations with finite fourth-order moments... |

9 |
Asymptotic behavior of a class of confidence regions based on ranks in regression
- Koul
- 1971
(Show Context)
Citation Context |

9 |
Multivariate rank tests
- Marden
- 1999
(Show Context)
Citation Context ...ign- and rank-based competitors of the Gaussian likelihood procedures are proposed. The ranks used by the authors are the spatial ranks introduced by Möttönen and Oja [37] and Chaudhuri [9]; see also =-=[32]-=-. Although Pitman efficiencies (with respect to the Gaussian methods) are obtained, no attempt is made to achieve any optimality and the authors restrict themselves to procedures of the Wilcoxon and s... |

9 | Influence function and asymptotic efficiency of scatter matrix based array processors: Case
- Ollila, Koivunen
- 2009
(Show Context)
Citation Context .../2 [ Ik2 +Kk − 2 k Jk ] (V ⊗2 ) −1/2 M ′ k =: Jk(f1)Υ −1 k (V), a form that is not unfamiliar in the area of robust estimation of covariance matrices; see, for instance, the asymptotic covariances in =-=[40, 42, 50, 51]-=- for the covariances of scatter estimates [as in (2.6), (2.7)], [41, 52] for covariances of shape estimates [as in (3.2)]. In the sequel, optimality (in the local and asymptotic sense, at radial densi... |

8 |
Rank-based robust analysis of linear models (with discussion),
- Draper
- 1988
(Show Context)
Citation Context ..., is by no means straightforward. Traditional R-estimators are defined (and computed) via the minimization of some rank-based objective function; see [1, 19, 20, 24, 26] or the review paper by Draper =-=[6]-=-. In the present context, this approach, in connection with (1.1), leads to the definition of an R-estimator as (1.2) (n) V := argmin Q ∼f1 f1 V ∼ ( (V) = argmin tr(S V 2 f1 ) 1 (V)) − (trSf1(V))2 , k... |

8 |
On a characterization of the normal distribution
- Kac
- 1939
(Show Context)
Citation Context ...nd identically distributed (i.i.d.), with common unspecified symmetric marginal density f. Under Gaussian marginals, i.i.d.-ness and sphericity coincide, but not under general densities. In fact, Kac =-=[27]-=- shows that this hypothesis of i.i.d.-ness is rotation-invariant only on the class of multivariate Gaussian distributions. If Gaussian assumptions are abandoned, this hypothesis is no longer rotation-... |

8 | Theory of Rank Tests, 2nd edition - Sidák, Sen - 1999 |

7 |
A linearized version of the Hodges-Lehmann estimator
- Antille
- 1974
(Show Context)
Citation Context ...g the arbitrary choice of a confidence level (1 −α) which has quite an impact on the final result. Another simple method can be obtained from the asymptotic linearity property of rank statistics (see =-=[2, 29]-=- or [25], page 321 for univariate location and regression). This method extends quite easily to the present context via the asymptotic linearity property (2.9). The latter indeed implies that for all ... |

7 |
Linearized rank estimates and signed-rank estimates for the general linear hypothesis
- Eeden, Kraft
- 1972
(Show Context)
Citation Context ...g the arbitrary choice of a confidence level (1 −α) which has quite an impact on the final result. Another simple method can be obtained from the asymptotic linearity property of rank statistics (see =-=[2, 29]-=- or [25], page 321 for univariate location and regression). This method extends quite easily to the present context via the asymptotic linearity property (2.9). The latter indeed implies that for all ... |

7 |
On the centering of a simple linear rank statistic
- Hoeffding
- 1973
(Show Context)
Citation Context ...(5.4) i i (ii) If the square integrability condition on K1 and K2 in (i) is reinforced J(Ki) := ∫ 1 0 u 1/2 (1 − u) 1/2 dKi(u) < ∞, i = 1,2 is consis(a classical condition that goes back to Hoeffding =-=[22]-=-), then φ ˜ tent iff, under K (n)(f), as n → ∞, n −1/2 n∑ [ ( n∑ ) ] 1 ∣∣∣U (n) E K P[dj ≤ di|di,Ui,Uj] n i=1 j=1 (5.5) ( × vec UiU ′ i − 1 k Ik ) P −→ ∞. (n) K (iii) If the Hoeffding condition (5.4) ... |

7 | The Affine equivariant sign covariance matrix: asymptotic behavior and efficiencies
- Ollila, Oja, et al.
- 2003
(Show Context)
Citation Context .../2 [ Ik2 +Kk − 2 k Jk ] (V ⊗2 ) −1/2 M ′ k =: Jk(f1)Υ −1 k (V), a form that is not unfamiliar in the area of robust estimation of covariance matrices; see, for instance, the asymptotic covariances in =-=[40, 42, 50, 51]-=- for the covariances of scatter estimates [as in (2.6), (2.7)], [41, 52] for covariances of shape estimates [as in (3.2)]. In the sequel, optimality (in the local and asymptotic sense, at radial densi... |

6 |
Estimating integrated squared density derivatives
- Bickel, Ritov
- 1988
(Show Context)
Citation Context ... involve a kernel estimate of g1 and, hence, cannot be expected to perform well under small and moderate sample sizes. Such kernel methods have been considered, for Wilcoxon scores, by [41] (see also =-=[3, 4, 7]-=- and, in a more general setting, in Section 4.5 of [27]. They also require arbitrary choices (window width and kernel or, as in [27], the choice of the order α of an empirical quantile) for which univ... |

6 |
On estimation of a class of efficiency-related parameters
- Cheng, Serfling
- 1981
(Show Context)
Citation Context ... involve a kernel estimate of g1 and, hence, cannot be expected to perform well under small and moderate sample sizes. Such kernel methods have been considered, for Wilcoxon scores, by [41] (see also =-=[3, 4, 7]-=- and, in a more general setting, in Section 4.5 of [27]. They also require arbitrary choices (window width and kernel or, as in [27], the choice of the order α of an empirical quantile) for which univ... |

6 |
2006b). Parametric and semiparametric inference for shape: the role of the scale functional
- Hallin, Paindaveine
(Show Context)
Citation Context ...d that, whereas this loss depends on the definition of shape (that is, on the arbitrary choice of the normalization V11 = 1), the semiparametric information bound does not; see Sections HP3.1, HP3.2, =-=[14]-=- and [39] for details. A general result by Hallin and Werker [17] suggests that, in case (i) for all f1 ∈ FA and σ > 0, the sequence of parametric subexperiments P (n) σ2 [see (2.2)] is ULAN with cent... |

6 |
An estimator of the scale parameter for the rank analysis of linear models under general score functions, Scandinavian
- Koul, L, et al.
- 1987
(Show Context)
Citation Context ...of [27]. They also require arbitrary choices (window width and kernel or, as in [27], the choice of the order α of an empirical quantile) for which universal recommendation seems hardly possible (see =-=[28]-=- for an empirical investigation). Moreover, estimating the actual underlying density is somewhat incompatible with the group-invariance spirit of the rank-based approach: if, indeed, the unknown densi... |

6 | Asymptotic linearity of serial and nonserial multivariate signed rank statistics
- HALLIN, PAINDAVEINE
- 2005
(Show Context)
Citation Context ...ons, respectively. Indeed, these bounds can be obtained, for example, by letting η → 0 and η → ∞, respectively, in the power-exponential family of distributions considered in Section 1.2. We refer to =-=[18]-=- for more general results on efficiency losses in the related problem of estimating the shape parameter. Some numerical values of those relative losses (3.6) are provided in Table 1 where we consider:... |

5 | An Alternative Asymptotic Analysis of ResidualBased Statistics”,
- Andreou, Werker
- 2005
(Show Context)
Citation Context ...nder P (n) θ,σ2 requires more stringent ,V;g1 asymptotic linearity results, such as those in Proposition A.1 of [16], or more general methods, such as the one recently developed by Andreou and Werker =-=[2]-=-. (n) Note that Q K; ˜ ˆ is no longer strictly invariant or distribution-free, but reθ mains asymptotically so, in the sense of being asymptotically equivalent to (n) its genuinely invariant and distr... |

5 |
Testing for ellipsoidal symmetry of a multivariate distribution
- Koltchinskii, Sakhanenko
- 2000
(Show Context)
Citation Context ...including the nonelliptical ones. The drawback is that they are computationally heavy and only achieve slow nonparametric consistency rates. Examples include Beran [6] and Koltchinskii and Sakhanenko =-=[28]-=- for the null hypothesis of ellipticity and Baringhaus [5] for sphericity. Another way of escaping Gaussian or fourth-order moment assumptions involves basing the tests on statistics that are measurab... |

4 | Serial and Nonserial Sign-and-rank Statistics: Asymptotic Representation and Asymptotic Normality”. The Annals of Statistics
- Hallin, Vermandele, et al.
- 2006
(Show Context)
Citation Context ...V ⊗2 ) −1 n 1/2 vec(V (n) # + oP(1) −V) = oP(1), − V) (still under P (n) σ2 , as n → ∞) and since the square-integrability of Kf1 ,V;g1 over (0,1) implies that m (n) f1 proof of Proposition 3.2(i) in =-=[16]-=-). − k = m(n) f1 − ∫ 1 0 Kf1 (u)du = o(n−1/2 ) (see theEFFICIENT RANK-BASED INFERENCE FOR SHAPE II 33 (n) Now, denoting by V ∼ (n) f1# := V ∼f1# (c0) the pseudo-estimator defined in (3.1), (f1,g1)] i... |

4 |
Nonparametric estimate of regression coefficients
- ˇCKOVÁ, J
- 1971
(Show Context)
Citation Context ...tion. The derivation of such R-estimators, however, is by no means straightforward. Traditional R-estimators are defined (and computed) via the minimization of some rank-based objective function; see =-=[1, 19, 20, 24, 26]-=- or the review paper by Draper [6]. In the present context, this approach, in connection with (1.1), leads to the definition of an R-estimator as V∼ (n) f1 := argmin V Q∼f1(V) = argminV ( tr(S2f1(V))−... |

3 |
Robustness and efficiency of scatter matrices
- TYLER
- 1983
(Show Context)
Citation Context ...ivariate setting, as the asymptotic covariance matrices of (the vec versions of) ̂ V ∼ Qk(V). Their relative performances can thus be described by a single number, a fact that was already observed in =-=[44]-=- (see also [33]); the situation is entirely different for covariance matrices, where two numbers are required [36, 37, 43]. These ARE’s coincide with those obtained in HP for the problem of testing V ... |

3 | Testing for proportionality of multivariate dispersion structures using interdirections
- Ghosh, Sengupta
- 2001
(Show Context)
Citation Context ...ultivariate concepts of signs and ranks developed, mainly, in the robustness literature; see [39] for a review. This sign-and/or-rank-based approach has been adopted by Tyler [53], Ghosh and Sengupta =-=[13]-=- and Marden and Gao [33]. Tyler [53] addresses the problem of testing uniformity over the sphere for directional data and proposes a sign test related to his celebrated [52] estimator of shape. In a s... |

3 | Rank-based procedures for structural hypotheses on covariance matrices
- Marden, Gao
- 2002
(Show Context)
Citation Context ...signs and ranks developed, mainly, in the robustness literature; see [39] for a review. This sign-and/or-rank-based approach has been adopted by Tyler [53], Ghosh and Sengupta [13] and Marden and Gao =-=[33]-=-. Tyler [53] addresses the problem of testing uniformity over the sphere for directional data and proposes a sign test related to his celebrated [52] estimator of shape. In a slightly different contex... |

3 |
Asymptotic linearity of a rank statistic in regression
- ˇCKOVÁ, J
- 1969
(Show Context)
Citation Context ... as in part (i), except that dni and Uni replace d0i and U0i , respectively. Accordingly, part (ii) holds under P(n) θn,σ 2,Vn;g1 . That it also holds under P (n) θ,σ 2,V;g1 follows from Lemma 3.5 of =-=[23]-=-. (iii) Note that |Jk(f1, g1)−J ( )k (f1;g1)|2 = | ∫ 1 0 (Kf1(u)−K( )f1 (u))Kg1(u) du|2 ≤ Jk(g1) ∫ 10 |Kf1(u) − K( )f1 (u)|2 du. Again, |K( )f1 (u) − Kf1(u)|2 ≤ 4|Kf1(u)|2 with ∫ 1 0 |Kf1(u)|2 du < ∞.... |

3 | Werker (2003). Semiparametric efficiency, distributionfreeness, and invariance - Hallin, M |

2 |
Testing for elliptical symmetry of a multivariate density
- Beran
- 1979
(Show Context)
Citation Context ...possible nonspherical alternatives, including the nonelliptical ones. The drawback is that they are computationally heavy and only achieve slow nonparametric consistency rates. Examples include Beran =-=[6]-=- and Koltchinskii and Sakhanenko [28] for the null hypothesis of ellipticity and Baringhaus [5] for sphericity. Another way of escaping Gaussian or fourth-order moment assumptions involves basing the ... |

2 |
The sample covariance is not efficient for elliptical distributions
- Falk
- 2002
(Show Context)
Citation Context ... This LAN (ULAN) issue has been briefly touched by f1 Bickel (Example 4 in [7]). The very particular case of bivariate distributions with finite second-order moments has been treated recently by Falk =-=[11]-=- in his investigation of the inefficiency of empirical correlation coefficients. In order to describe the extremely mild assumptions under which the families P (n) are ULAN, we introduce the following... |

2 |
Validity conditions in repeated-measures designs
- Huynh, Mandeville
- 1979
(Show Context)
Citation Context ...ithout elliptical symmetry, however, these adjusted tests are no longer valid; therefore, they qualify as tests of sphericity, not as tests of isotropy or unit shape. Moreover, it has been shown (see =-=[23]-=-) that they behave rather badly under heavy tails (a fact that is confirmed by the Monte Carlo study in Section 6). Although they still require elliptical symmetry and finite fourthorder radial moment... |

2 |
Robust Statistical Procedures: Asymptotics and Interrelations
- ˇCKOVÁ, J, et al.
- 1996
(Show Context)
Citation Context ...ary choice of a confidence level (1 − α) which has quite an impact on the final result. Another simple method can be obtained from the asymptotic linearity property of rank statistics (see [2, 29] or =-=[25]-=-, page 321 for univariate location and regression). This method extends quite easily to the present context via the asymptotic linearity property (2.9). The latter indeed implies that for all f1, g1 ∈... |

2 | Werker (2004). An alternative asymptotic analysis of residual-based statistics - Andreou, M |

2 | Yang (2000). Asymptotics in Statistics, 2nd edition - Cam, L |

1 |
Affine-equivariant R-estimation of shape. Manuscript in preparation
- HALLIN, OJA, et al.
- 2006
(Show Context)
Citation Context ...to [15], where it is shown that an adequate modification of ̂ (n) V f1 ∼ producing a strictly equivariant ̂ (n) V f1 is possible (at ∼ the price of some technicalities). Alternatively, it is shown in =-=[8]-=- that, under mild additional assumptions, an affine-equivariant R-estimator of shape also (n) can be obtained from iterating the mapping V ↦→ W ∼ f1 (V)/(W (n) ∼ f1 (V))11, (n) where W is defined in (... |

1 |
Affine-equivariant multivariate sign methods
- Ollila, Hettmansperger, et al.
- 2004
(Show Context)
Citation Context ...sistency requirement (i), such as Tyler’s [45] celebrated affine-equivariant estimator of shape (scatter, in Tyler’s terminology) V (n) T or the estimator of shape based on the Oja signs developed in =-=[36]-=-. These estimators are indeed root-n consistent under extremely general conditions (second-order moments, however, are required in [36]), but they are not efficient. The answer, as we shall see, is po... |

1 |
The affine-equivariant sign covariance matrix: asymptotic behaviour and efficiencies
- Ollila, Oja, et al.
- 2003
(Show Context)
Citation Context ...rmances can thus be described by a single number, a fact that was already observed in [44] (see also [33]); the situation is entirely different for covariance matrices, where two numbers are required =-=[36, 37, 43]-=-. These ARE’s coincide with those obtained in HP for the problem of testing V = V0 (see Proposition HP4.2). An immediate corollary is that the Chernoff–Savage result of [38] also applies here: the ARE... |

1 |
A canonical definition of shape, submitted
- Paindaveine
- 2006
(Show Context)
Citation Context ...hereas this loss depends on the definition of shape (that is, on the arbitrary choice of the normalization V11 = 1), the semiparametric information bound does not; see Sections HP3.1, HP3.2, [14] and =-=[39]-=- for details. A general result by Hallin and Werker [17] suggests that, in case (i) for all f1 ∈ FA and σ > 0, the sequence of parametric subexperiments P (n) σ2 [see (2.2)] is ULAN with central seque... |

1 | Paindaveine (2006a). Asymptotic linearity of serial and nonserial multivariate signed rank statistics - Hallin, D |

1 | Paindaveine (2006b). On parametrically and semiparametrically efficient estimation of shape in elliptic distributions - Hallin, D |