SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE I. OPTIMAL RANK-BASED TESTS FOR SPHERICITY

By Marc Hallin, Davy Paindaveine. 2006

143 citations recorded by the source corpus.

We propose a class of rank-based procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value V 0 ; this includes, for V 0 = I k , the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so when it must be estimated. The multivariate ranks used throughout are those of the distances-in the metric associated with the null value V 0 of the shape matrix-between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31-43]) or, equivalently, with respect to (an extension of) the test for sphericity introduced by John [Biometrika 59 (1972) 169-173]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performance is investigated via a Monte Carlo study.

Subfields: Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4eac403b3c2c905faedc5ecfb86147e2d9bc55a6

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Citations in the corpus

Monge-Kantorovich Depth, Quantiles, Ranks and Signs (2014, background): “There, spherical contours with Pμ,I,g-probability contents τ coincide with the halfspace depth contours, and provide a natural definition of τ -quantile contours for Y , while RP (Y ), RP (Y )/‖RP (Y )‖ and ‖RP (Y )‖ play the roles of vector ranks, signs, and ranks, respectively ([21, 22, 23, 24, 25”
Multivariate sign-based high-dimensional tests for sphericity (2014, methodology): “See Hallin & Paindaveine (2006) for the proof.”
ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions (2013, background): “Theoretical analysis in low dimensions is provided by Hallin and Paindaveine (2002a, 2002b, 2004, 2005, 2006), Hallin (Oja); Hallin, Paindaveine, and Verdebout (2010, 2014), and some new extensions to high-dimensional settings are provided in Croux, Filzmoser, and Fritz (2013) and Feng and Liu…”
OPTIMAL RANK-BASED TESTING FOR PRINCIPAL COMPONENTS (2010, methodology): “LAN (actually, ULAN) has been established, under appropriate regularity assumptions on radial densities, in Hallin and Paindaveine (2006a), for elliptical families when parametrized by a location vector θ and a scatter matrix [more precisely, the vector vech( ) resulting from stacking the upper…”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, methodology): “…linear models with VARMA errors (Hallin and Paindaveine 2004a, 2005a, 2006a), VAR order identification (Hallin and Paindaveine 2004b), shape (Hallin and Paindaveine 2006b; Hallin, Oja, and Paindaveine 2006), homogeneity of scatter (Hallin and Paindaveine 2008), principal and common…”
Multivariate-Sign-Based High-Dimensional Tests for the Two-Sample Location Problem (2016, methodology): “Besides the spatial-sign-based sphericity test, there are some other tests based on spatial-signed-rank or spatial-rank in the literatures, such as Hallin and Paindaveine (2006).”
Asymptotic power of sphericity tests for high-dimensional data (2013, methodology): “…scale and orthogonal transformations) test was obtained by John (1971, 1972) and by Sugiura (1972), with adjusted versions resisting elliptical violations of the Gaussian assumptions proposed in Hallin and Paindaveine (2006), where a Le Cam approach is adopted under a general elliptical setting.”
SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE II. OPTIMAL R-ESTIMATION OF SHAPE (2006, unknown): “2 of [13].”
Invariant co‐ordinate selection (2009, background): “This question, quite presumably, is related to the problem of constructing 'optimal' tests for sphericity (robust alternatives to the traditional Mauchly (1940) and John (1972) tests can be found in Tyler (1982, 1987) and Hallin and Paindaveine (2006)).”
On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions (2017, methodology): “…T E(n) θ,g,Λ } rejects the null whenever Qscθ = ( ∆ T E(n) θ;2 )T ( ΓT Eθ;22 )−1 ∆ T E(n) θ;2 > χ 2 (p−2)(p+1)/2,1−α. (8) Now, using the closed form for the inverse of ΓT Eθ;22 in Lemma 5.2 from Hallin and Paindaveine (2006a), it is easy to show that the test statistic in (8) coincides with Qscθ .”
A canonical definition of shape (2008, unknown): “…nonspecification of σ2S does not cause any loss of efficiency when performing inference on VS (the loss of efficiency, for any fixed S, is studied in Hallin and Paindaveine 2006b, both for point estimation and hypothesis testing), hence, for which inference on VS in PnS and Pn S;σ2 S ,f1…”