A New Sobolev Test for Uniformity on the Circle

Hermans, M.; Rasson, Jean‐Paul

doi:10.2307/2336748

Cited by 6 publications

(9 citation statements)

References 2 publications

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“…Hermans & Rasson (1985) use a general method for building Beran's type tests for uniformity which are modifications of Ajne's statistic and designed to be omnibus and powerful against the alternatives $H_{1,\ell}$ , $H_{2,\ell}$ , and $H_{4,\ell}$ . They introduce a family of new statistics indexed by a real parameter, among which $R_n$ ($T_{n,\beta_2}$ in their paper) seems to be the most convenient choice.…”

Section: Beran's Class Of Tests Of Uniformitymentioning

confidence: 99%

“…In Table 5 are given the empirical powers of the tests computed from 100,000 samples of size $n=20$ for the cases $(m,\ell)\in\{1,2,3,4\}\times \{1.2,1.5,2,3,4\}$ . These values were already used by Stephens (1969) and Hermans & Rasson (1985), except the case $m=3$ . For $U_n^2$ , $A_n$ and $R_n$ our values are very close to those given in Table 4 by Stephens (1969) and reproduced in Hermans & Rasson (1985) in Table 2.…”

Section: Power Comparisons Under Stephens's Alternativesmentioning

confidence: 99%

“…These values were already used by Stephens (1969) and Hermans & Rasson (1985), except the case $m=3$ . For $U_n^2$ , $A_n$ and $R_n$ our values are very close to those given in Table 4 by Stephens (1969) and reproduced in Hermans & Rasson (1985) in Table 2. The main conclusions from our simulations are the following.…”

Section: Power Comparisons Under Stephens's Alternativesmentioning

confidence: 99%

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Some tests for uniformity of circular distributions powerful against multimodal alternatives

Pycke

2010

Can J Statistics

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The author introduces new statistics suited for testing uniformity of circular distributions and powerful against multimodal alternatives. One of them has a simple expression in terms of the geometric mean of the sample of chord lengths. The others belong to a family indexed by a continuous parameter. The asymptotic distributions under the null hypothesis are derived. We compare the power of the new tests against Stephens's alternatives with those of Ajne, Watson, and Hermans‐Rasson's tests. Some of the new tests are the most powerful when the alternative has three or four modes. A heuristic justification of this feature is given. An application to the analysis of archaeological data is provided. The Canadian Journal of Statistics 38:80–96; 2010 © 2010 Statistical Society of Canada

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Section: Beran's Class Of Tests Of Uniformitymentioning

confidence: 99%

Section: Power Comparisons Under Stephens's Alternativesmentioning

confidence: 99%

Section: Power Comparisons Under Stephens's Alternativesmentioning

confidence: 99%

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Some tests for uniformity of circular distributions powerful against multimodal alternatives

Pycke

2010

Can J Statistics

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“…The investigation that has been undertaken (Stephens 1969 ; Bergin 1991 ; Bogdan et al 2002 ; Pycke 2010 ) has highlighted potential for low power (as discussed above). For this reason, alternative tests have been proposed that were designed specifically to have better power against multimodal alternatives: the two most developed that we could find in the literature were due to Hermans and Rasson ( 1985 ) and Bogdan et al ( 2002 ); see Pycke ( 2010 ) for theoretical underpinning of general classes to tests for circular uniformity. Both have been subject to very little published evaluation, and neither is available in any software package that we know of.…”

Section: Introductionmentioning

confidence: 99%

Circular data in biology: advice for effectively implementing statistical procedures

Landler

Ruxton

Malkemper

2018

Behav Ecol Sociobiol

134

120

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Circular data are common in biological studies. The most fundamental question that can be asked of a sample of circular data is whether it suggests that the underlying population is uniformly distributed around the circle, or whether it is concentrated around at least one preferred direction (e.g. a migratory goal or activity phase). We compared the statistical power of five commonly used tests (the Rayleigh test, the V-test, Watson’s test, Kuiper’s test and Rao’s spacing test) across a range of different unimodal scenarios. The V-test showed higher power for symmetrical distributions, Rao’s spacing performed worst for all explored unimodal distributions tested and the remaining three tests showed very similar performance. However, the V-test only applies if the hypothesis is restricted to one (pre-specified) direction of interest. In all other unimodal cases, we recommend using the Rayleigh test. Much less explored is the multimodal case with data concentrated around several directions. We performed power simulations for a variety of multimodal situations, testing the performance of the widely used Rayleigh, Rao’s, Watson, and Kuiper’s tests as well as the more recent Bogdan and Hermans-Rasson tests. Our analyses of alternative statistical methods show that the commonly used tests lack statistical power in many of multimodal cases. Transformation of the raw data (e.g. doubling the angles) can overcome some of the issues, but only in the case of perfect f-fold symmetry. However, the Hermans-Rasson method, which is not yet implemented in any software package, outcompetes the alternative tests (often by substantial margins) in most of the multimodal situations explored. We recommend the wider uptake of the powerful but hitherto neglected Hermans-Rasson method. In summary, we provide guidance for biologists helping them to make decisions when testing circular data for single or multiple departures from uniformity.Electronic supplementary materialThe online version of this article (10.1007/s00265-018-2538-y) contains supplementary material, which is available to authorized users.

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“…Particular attention is devoted to p = 2 (circular data) and p = 3 (spherical data). In these cases the asymptotic behavior of proposed tests under the null hypothesis is established using two approaches: first is based on an adaptation of methods of goodness of fit tests described in [1,2], and second using Gine theory based on Sobolev norms [9,10].…”

Section: Introductionmentioning

confidence: 99%