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Adaptive tests for ANOVA in Fisher–von Mises–Langevin populations under heteroscedasticity

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Abstract

Fisher–von Mises–Langevin distributions are widely used for modeling directional data. In this paper, the problem of testing homogeneity of mean directions of several Fisher–von Mises–Langevin populations is considered when the concentration parameters are unknown and heterogeneous. First, an adaptive test based on the likelihood ratio statistic is proposed. Critical points are evaluated using a parametric bootstrap. Second, a heuristic test statistic is considered based on pairwise group differences. A nonparametric bootstrap procedure is adapted for evaluating critical points. Finally, a permutation test is also proposed. An extensive simulation study is performed to compare the size and power values of these tests with those proposed earlier. It is observed that both parametric and nonparametric bootstrap based tests achieve size values quite close to the nominal size. Asymptotic tests and permutation tests have size values higher than the nominal size. Bootstrap tests are seen to have very good power performance. The robustness of tests is also studied by considering contamination in Fisher–von Mises–Langevin distributions. R packages are developed for the actual implementation of all tests. A real data set has been considered for illustrations.

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References

  • Abramowitz M, Stegun IA, Romer RH (1988) Handbook of mathematical functions with formulas. Graphs and mathematical tables. American Association of Physics Teachers, College Park

    Book  Google Scholar 

  • Abzhanov A, Kallemov B (2013) Using von Mises–Fisher distribution for polymer conformation analysis in multi-scale framework. Procedia Eng 61:111–116

    Article  Google Scholar 

  • Amaral GJA, Dryden IL, Wood ATA (2007) Pivotal bootstrap methods for k-sample problems in directional statistics and shape analysis. J Am Stat Assoc 102(478):695–707

    Article  MathSciNet  CAS  Google Scholar 

  • Banerjee A, Dhillon IS, Ghosh J, Sra S, Ridgeway G (2005) Clustering on the unit hypersphere using von Mises–Fisher distributions. J Mach Learn Res 6(9):1

    MathSciNet  Google Scholar 

  • Chung E, Romano JP (2016) Multivariate and multiple permutation tests. J Econ 193(1):76–91

    Article  MathSciNet  Google Scholar 

  • Figueiredo A (2012) Goodness-of-fit for a concentrated von Mises–Fisher distribution. Comput Stat 27(1):69–82

    Article  MathSciNet  Google Scholar 

  • Figueiredo A (2017) Bootstrap and permutation tests in ANOVA for directional data. Comput Stat 32(4):1213–1240

    Article  MathSciNet  Google Scholar 

  • Figueiredo A (2021) Resampling methods in ANOVA for data from the von Mises–Fisher distribution. Commun Stat-Simul Comput. https://doi.org/10.1080/03610918.2021.1976796

    Article  Google Scholar 

  • Fisher NI, Lewis T, Embleton BJJ (1987) Statistical analysis of spherical data. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Friedrich S, Pauly M (2018) MATS: inference for potentially singular and heteroscedastic MANOVA. J Multivar Anal 165:166–179

    Article  MathSciNet  Google Scholar 

  • Friedrich S, Brunner E, Pauly M (2017) Permuting longitudinal data in spite of the dependencies. J Multivar Anal 153:255–265

    Article  MathSciNet  Google Scholar 

  • Hornik K, Grun B (2014) movMF: an R package for fitting mixtures of von Mises–Fisher distributions. J Stat Softw 58(10):1–31

    Article  Google Scholar 

  • Jammalamadaka SR, SenGupta A (2001) Topics in circular statistics. World Scientific, Singapore

    Book  Google Scholar 

  • Johnson EN, Richter SJ (2022) Permutation tests for mixed paired and two-sample designs. Comput Stat 37(2):739–750

    Article  MathSciNet  Google Scholar 

  • Kanika, Kumar S (2019) On efficiency and robustness of estimators for a spherical location. Statistics 53(3):601–629

    Article  MathSciNet  Google Scholar 

  • Kanika, Kumar S, SenGupta A (2015) A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions. J Multivar Anal 133:160–172

    Article  MathSciNet  Google Scholar 

  • Konietschke F, Bathke AC, Harrar SW, Pauly M (2015) Parametric and nonparametric bootstrap methods for general MANOVA. J Multivar Anal 140:291–301

    Article  MathSciNet  Google Scholar 

  • Kulkarni HV, SenGupta A (2022) An efficient test for homogeneity of mean directions on the hyper-sphere. Int Stat Rev 90(1):41–61

    Article  MathSciNet  Google Scholar 

  • Langevin P (1905) Magnétisme et théorie des électrons. Ann Chim Phys 8:203

    Google Scholar 

  • Ley C, Verdebout T (2017) Modern directional statistics. CRC, Boca Raton

    Book  Google Scholar 

  • Ley C, Verdebout T (2018) Applied directional statistics: modern methods and case studies. CRC Press, Boca RAton

    Book  Google Scholar 

  • Ley C, Swan Y, Verdebout T (2017) Efficient ANOVA for directional data. Ann Inst Stat Math 69(1):39–62

    Article  MathSciNet  Google Scholar 

  • Mardia KV, Jupp PE (2000) Directional statistics. Wiley, Chichester

    Google Scholar 

  • Noguchi K, Konietschke F, Marmolejo-Ramos F, Pauly M (2021) Permutation tests are robust and powerful at 0.5% and 5% significance levels. Behav Res Methods 53(6):2712–2724

    Article  PubMed  Google Scholar 

  • Pewsey A, García-Portugués E (2021) Recent advances in directional statistics. TEST 30(1):1–58

    Article  MathSciNet  Google Scholar 

  • Pewsey A, Neuhäuser M, Ruxton GD (2013) Circular statistics in R. Oxford University Press, Oxford

    Google Scholar 

  • Rumcheva P, Presnell B (2017) An improved test of equality of mean directions for the Langevin von Mises Fisher distribution. Aust N Z J Stat 59(1):119–135

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

  • Stephens MA (1972) Multisample tests for the von Mises distribution. J Am Stat Assoc 67(338):456–461

    Article  Google Scholar 

  • Stephens MA (1992) On Watson’s ANOVA for directions. Technical Report No. TR-454, Department of Statistics, Stanford Univeristy

  • Tasdan F, Yeniay O (2018) A comparative simulation of multiple testing procedures in circular data problems. J Appl Stat 45(2):255–269

    Article  MathSciNet  Google Scholar 

  • Upton GJG (1976) More multisample tests for the von Mises distribution. J Am Stat Assoc 71(355):675–678

    Article  Google Scholar 

  • Watson GS (1983) Statistics on spheres. Wiley, New York

    Google Scholar 

  • Watson GS, Debiche MG (1992) Testing whether two Fisher distributions have the same centre. Geophys J Int 109(1):225–232

    Article  ADS  Google Scholar 

  • Watson GS, Williams E (1956) On the construction of significance tests on the circle and the sphere. Biometrika 43(3):344–352

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the co-editor, an associate editor and the reviewers for their valuable suggestions which have led to substantial improvement in the content and presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Shreyashi Basak & Somesh Kumar

  2. Department of Statistics, TU Dortmund University, 44227, Dortmund, Germany

    Markus Pauly

  3. Research Center Trustworthy Data Science and Security, UA Ruhr, 44227, Dortmund, Germany

    Markus Pauly

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  1. Shreyashi Basak
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  2. Markus Pauly
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  3. Somesh Kumar
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Correspondence to Somesh Kumar.

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Basak, S., Pauly, M. & Kumar, S. Adaptive tests for ANOVA in Fisher–von Mises–Langevin populations under heteroscedasticity. Comput Stat 39, 433–459 (2024). https://doi.org/10.1007/s00180-022-01298-1

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  • DOI: https://doi.org/10.1007/s00180-022-01298-1

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