Abstract

In the present paper, we tackle the problem of detecting serial corre

lation in directional data. We introduce a concept of runs properly adapted to

the directional context. We then show that tests based on the latter runs enjoy

some local and asymptotic optimality properties against local alternatives with

serial dependence. We evaluate the finite-sample performances of our tests using

Monte Carlo simulations and show their usefulness on a real data illustration

that involves the analysis of sunspots locations for various solar cycles.

Information

Preprint No.SS-2024-0106
Manuscript IDSS-2024-0106
Complete AuthorsMaxime Boucher, Christian Francq, Yuichi Goto, Thomas Verdebout
Corresponding AuthorsThomas Verdebout
Emailstverdebout@gmail.com

References

  1. Accardi, L., J. Cabrera, G. Watson, et al. (1987). Some stationary markov processes in discrete time for unit vectors. Metron 45(1-2), 115–133.
  2. Alonso-Pena, M., J. Ameijeiras-Alonso, and R. M. Crujeiras (2021). Nonparametric tests for circular regression. Journal of Statistical Computation and Simulation 91(3), 477–500.
  3. Alonso-Pena, M., I. Gijbels, and R. M. Crujeiras (2024). A general framework for circular local likelihood regression. Journal of the American Statistical Association 119 (548), 2709–2721.
  4. Babcock, H. W. (1961). The topology of the Sun’s magnetic field and the 22-year cycle. Astrophysical Journal 133(2), 572–587.
  5. Baranyi, T., L. Gy˝ori, and A. Ludm´any (2016). On-line tools for solar data compiled at the Debrecen observatory and their extensions with the Greenwich sunspot data. Solar Physics 291(9), 3081–3102.
  6. Biswas, M., M. Mukhopadhyay, and A. K. Ghosh (2014). A distribution-free two-sample run test applicable to high-dimensional data. Biometrika 101(4), 913–926.
  7. Cho, J. S. and H. White (2011). Generalized runs tests for the iid hypothesis. Journal of Econometrics 162(2), 326–344.
  8. Corzo, J. and G. Babativa (2013). A modified runs test for symmetry. Journal of Statistical Computation and Simulation 83(5), 984–991.
  9. Di Marzio, M., S. Fensore, A. Panzera, and C. C. Taylor (2017). Nonparametric estimating equations for circular probability density functions and their derivatives. Electronic Journal of Statistics 11(2), 4323–4346.
  10. Dufour, J.-M., M. Hallin, and I. Mizera (1998). Generalized runs tests for heteroscedastic time series. Journal of Nonparametric Statistics 9(1), 39–86.
  11. Dyckerhoff, R., C. Ley, and D. Paindaveine (2015). Depth-based runs tests for bivariate central symmetry. Annals of the Institute of Statistical Mathematics 67, 917–941.
  12. Garc´ıa-Portugu´es, E., P. L. de Micheaux, S. G. Meintanis, and T. Verdebout (2024). Nonparametric tests of independence for circular data based on trigonometric moments. Statistica Sinica, 567–588.
  13. Garc´ıa-Portugu´es, E., D. Paindaveine, and T. Verdebout (2020). On optimal tests for rotational symmetry against new classes of hyperspherical distributions. Journal of the American Statistical Association 115(532), 1873–1887.
  14. Garc´ıa-Portugu´es, E., I. Van Keilegom, R. M. Crujeiras, and W. Gonz´alez-Manteiga (2016). Testing parametric models in linear-directional regression. Scandinavian Journal of Statistics 43(4), 1178–1191.
  15. Gy˝ori, L., T. Baranyi, and A. Lud´amny (2016). Comparative analysis of Debrecen sunspot catalogues. Monthly Notices of the Royal Astronomical Society 465(2), 1259–1273.
  16. Haigh, J. D. (2007). The Sun and the Earth’s climate. Living Reviews in Solar Physics 4, 2.
  17. Hallin, M., H. Liu, and T. Verdebout (2024). Nonparametric measure-transportation-based methods for directional data. Journal of the Royal Statistical Society, Series B 86 (5), 1172–1196.
  18. Hentati-Kaffel, R. and P. De Peretti (2015). Generalized runs tests to detect randomness in hedge funds returns. Journal of Banking and Finance 50, 608–615.
  19. Henze, N. and M. D. Penrose (1999). On the multivariate runs test. Annals of Statistics 27(1), 290–298.
  20. Ley, C., Y. Swan, B. Thiam, and T. Verdebout (2013). Optimal R-estimation of a spherical location. Statistica Sinica 23(1), 305–332.
  21. Ley, C. and T. Verdebout (2017). Modern Directional Statistics. Chapman & Hall/CRC Interdisciplinary Statistics Series. Boca Raton: CRC Press.
  22. Marden, J. (1999). Multivariate rank tests. Statistics textbooks and monographs 159, 401–432.
  23. Mardia, K. V. and P. E. Jupp (1999). Directional Statistics. Wiley Series in Probability and Statistics. Chichester: Wiley.
  24. McWilliams, T. P. (1990). A distribution-free test for symmetry based on a runs statistic. Journal of the American Statistical Association 85(412), 1130–1133.
  25. Meil´an-Vila, A., M. Francisco-Fern´andez, R. M. Crujeiras, and A. Panzera (2020). Nonparametric multiple regression estimation for circular responses. Test 30(3)(650–672), to appear.
  26. Paindaveine, D. (2009). On multivariate runs tests for randomness. Journal of the American Statistical Association 104(488), 1525–1538.
  27. Paindaveine, D. and T. Verdebout (2016). On high-dimensional sign tests. Bernoulli 22(3), 1745–1769.
  28. Paindaveine, D. and T. Verdebout (2017). Inference on the mode of weak directional signals: a Le Cam perspective on hypothesis testing near singularities. Annals of Statistics 45(2), 800–832.
  29. Rao, J. and A. SenGupta (2001). Topics in Circular Statistics, Volume 5 of Series on Multivariate Analysis. Singapore: World Scientific.
  30. Schach, S. (1969). Nonparametric symmetry tests for circular distributions. Biometrika 56(3), 571–577.
  31. Solanki, S. K., B. Inhester, and M. Sch¨ussler (2006). The solar magnetic field. Reports on Progress in Physics 69(3), 563–668.
  32. Wald, A. and J. Wolfowitz (1940). On a test whether two samples are from the same population. The Annals of Mathematical Statistics 11(2), 147–162. Maxime Boucher D´epartement de Math´ematique and ECARES Universit´e libre de Bruxelles (ULB) Campus Plaine, Boulevard du Triomphe, CP210 B-1050 Brussels, Belgium

Acknowledgments

Maxime Boucher gratefully acknowledges the PDR grant 40008027 from the

Fonds National de la Recherche Scientifique (FNRS), Belgium, Christian

Francq gratefully acknowledges the support of the Agence Nationale de la

Recherche (ANR) through the Project MLEforRisk (ANR-21-CE26-0007),

Yuichi Goto gratefully acknowledges the JSPS Grant-in-Aid for Early-Career

Scientists JP23K16851 (Y.G.) while Thomas Verdebout gratefully acknowledges the PDR grant 40008027 from the Fonds National de la Recherche

Scientifique (FNRS), Belgium and an advanced ARC grand from the Communaut´e Fran¸caise de Belgique .

Supplementary Materials

The online supplementary material contains all the technical proofs and

complements to the real data illustration.

Supplementary materials are available for download.