An overview of uniformity tests on the hypersphere

García-Portugués, Eduardo; Verdebout, Thomas

doi:10.48550/arxiv.1804.00286

Cited by 6 publications

(8 citation statements)

References 26 publications

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“…n has several neat connections with other circular and linear uniformity tests; particularly it can be regarded as the rotation-invariant version of the CvM statistic that selects the origin in such a way that the discrepancy of the sample with respect to H 0 is minimized (see, e.g., García-Portugués and Verdebout (2018)). The relation of P CvM n,1 and U 2 n stems from Proposition 4 and the following alternative form for U 2 n (see, e.g., Mardia and Jupp (1999, page 111)):…”

Section: Extending the Watson Testmentioning

confidence: 99%

“…Since the second half of the 20th century, a sizeable number of tests for assessing uniformity on Ω q have been proposed. These contributions range notably in generality (arbitrary dimension vs. specific-dimension tests; consistency against all kind of deviations vs. consistency only against certain alternatives) and underlying methodology (parametric vs. nonparametric tests); see García-Portugués and Verdebout (2018) for an updated review. In addition to its self-importance, uniformity tests on Ω q are important auxiliary tools for, among others, the following statistical problems: (i) testing for spherically-symmetric distributions on R q+1 (see, e.g., Cai et al (2013)); (ii) goodnessof-fit tests on Ω 1 via the probability integral transform (Mardia and Jupp, 1999, Section 6.4.2); (iii) goodness-of-fit tests on Ω q , q ≥ 1, via an almost-canonical transformation (Jupp and Kume, 2020, Proposition 1); (iv ) testing for rotational symmetry on Ω q+1 (see, e.g., García-Portugués et al (2020a)).…”

Section: Introductionmentioning

confidence: 99%

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On a projection-based class of uniformity tests on the hypersphere

García‐Portugués¹,

Navarro-Esteban²,

Cuesta-Albertos³

2020

Preprint

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We propose a projection-based class of uniformity tests on the hypersphere using statistics that integrate, along all possible directions, the weighted quadratic discrepancy between the empirical cumulative distribution function of the projected data and the projected uniform distribution. Simple expressions for several test statistics are obtained for the circle and sphere, and relatively tractable forms for higher dimensions. Despite its different origin, the proposed class is shown to be related with the well-studied Sobolev class of uniformity tests. Our new class proves itself advantageous by allowing to derive new tests for hyperspherical data that neatly extend the circular tests by Watson, Ajne, and Rothman, and by introducing the first instance of an Anderson-Darling-like test for such data. The asymptotic distributions and the local optimality against certain alternatives of the new tests are obtained. A simulation study evaluates the theoretical findings and evidences that, for certain scenarios, the new tests are competitive against previous proposals. The new tests are employed in three astronomical applications.

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Section: Extending the Watson Testmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a projection-based class of uniformity tests on the hypersphere

García‐Portugués¹,

Navarro-Esteban²,

Cuesta-Albertos³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

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“…This is a very classical problem in multivariate analysis that can be traced back to Bernoulli (1735). As explained in the review paper García-Portugués and Verdebout (2018), the topic has recently received a lot of attention: to cite only a few contributions, Jupp (2008) proposed data-driven Sobolev tests, Cuesta-Albertos, Cuevas and Fraiman (2009) and García-Portugués, Navarro-Esteban and Cuesta-Albertos (2019) proposed tests based on random projections, Lacour and Pham Ngoc (2014) studied the problem for noisy data, Paindaveine and Verdebout (2016) obtained the high-dimensional limiting behavior of some classical test statistics under the null hypothesis while García-Portugués, Paindaveine and Verdebout (2019) transformed some uniformity tests into tests of rotational symmetry.…”

Section: Introductionmentioning

confidence: 99%

On the power of axial tests of uniformity on spheres

Cutting¹,

Paindaveine²,

Verdebout³

2019

Preprint

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Testing uniformity on the p-dimensional unit sphere is arguably the most fundamental problem in directional statistics. In this paper, we consider this problem in the framework of axial data, that is, under the assumption that the n observations at hand are randomly drawn from a distribution that charges antipodal regions equally. More precisely, we focus on axial, rotationally symmetric, alternatives and first address the problem under which the direction θ θ θ of the corresponding symmetry axis is specified. In this setup, we obtain Le Cam optimal tests of uniformity, that are based on the sample covariance matrix (unlike their non-axial analogs, that are based on the sample average). For the more important unspecified-θ θ θ problem, some classical tests are available in the literature, but virtually nothing is known on their non-null behavior. We therefore study the non-null behavior of the celebrated Bingham test and of other tests that exploit the single-spiked nature of the considered alternatives. We perform Monte Carlo exercises to investigate the finite-sample behavior of our tests and to show their agreement with our asymptotic results.

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“…Inference for location in the vicinity of the uniformity case was considered in Verdebout (2017, 2020a). We refer to García-Portugués and Verdebout (2018) for a recent review of uniformity tests.…”

Section: Introductionmentioning

confidence: 99%

“…The Bayesian optimality of Sobolev tests has recently been studied in Sun and Lockhart (2019). We refer to García-Portugués and Verdebout (2018) for an overview of the various Sobolev tests that have been considered in the literature.…”

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confidence: 99%

On the power of Sobolev tests for isotropy under local rotationally symmetric alternatives

García‐Portugués¹,

Paindaveine²,

Verdebout³

2021

Preprint

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We consider one of the most classical problems in multivariate statistics, namely the problem of testing isotropy, or equivalently, the problem of testing uniformity on the unit hypersphere S p−1 of R p . Rather than restricting to tests that can detect specific types of alternatives only, we consider the broad class of Sobolev tests. While these tests are known to allow for omnibus testing of uniformity, their non-null behavior and consistency rates, unexpectedly, remain largely unexplored. To improve on this, we thoroughly study the local asymptotic powers of Sobolev tests under the most classical alternatives to uniformity, namely, under rotationally symmetric alternatives. We show in particular that the consistency rate of Sobolev tests does not only depend on the coefficients defining these tests but also on the derivatives of the underlying angular function at zero. For any Sobolev test and any rotationally symmetric alternative, we derive the consistency rate of the Sobolev test and determine the corresponding local asymptotic powers. We show that Sobolev tests with non-zero coefficients at odd (respectively, even) ranks only are blind (at any polynomial rate) to alternatives with angular functions whose kth-order derivatives at zero vanish for any k odd (respectively, even). Our asymptotic analysis requires investigating the non-standard behavior of random Chebyshev polynomials (for p = 2) and random Gegenbauer polynomials (for p ≥ 3) in the vicinity of the uniform distribution on S p−1 . Our non-standard asymptotic results are illustrated with Monte Carlo exercises.

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An overview of uniformity tests on the hypersphere

Cited by 6 publications

References 26 publications

On a projection-based class of uniformity tests on the hypersphere

On a projection-based class of uniformity tests on the hypersphere

On the power of axial tests of uniformity on spheres

On the power of Sobolev tests for isotropy under local rotationally symmetric alternatives

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