Kernel density estimation for circular data: a Fourier series-based plug-in approach for bandwidth selection

References

• Bai, Z.D., Rao, C.R., and Zhao, L.C. (1988), ‘Kernel Estimators of Density Function of Directional Data’, Journal of Multivariate Analysis, 27, 24–39.
   Web of Science ®Google Scholar
• Batschelet, E. (1981), Circular Statistics in Biology, London: Academic Press.
   Google Scholar
• Beran, R. (1979), ‘Exponential Models for Directional Data’, The Annals of Statistics, 7, 1162–1178.
   Web of Science ®Google Scholar
• Bosq, D., and Lecoutre, J.-P. (1987), Théorie de l'Estimation Fonctionnelle, Paris: Economica.
   Google Scholar
• Butzer, P.L., and Nessel, R.J. (1971), Fourier Analysis and Approximation. One-Dimensional Theory, Basel: Birkhauser.
   Google Scholar
• Chiu, S.-T. (1991), ‘Bandwidth Selection for Kernel Density Estimation’, The Annals of Statistics, 19, 1883–1905.
   Web of Science ®Google Scholar
• Deheuvels, P. (1977), ‘Estimation Non Paramétrique De La Densité Par Histogrammes Généralisés’, Revue de Statistique Appliquée, 25(3), 5–42.
   Google Scholar
• Deheuvels, P., and Hominal, P. (1980), ‘Estimation Automatique De La Densité’, Revue de Statistique Appliquée, 28(1), 25–55.
   Google Scholar
• Di Marzio, M, Panzera, A., and Taylor, C.C. (2009), ‘Local Polynomial Regression for Circular Predictors’, Statistics & Probability Letters, 79, 2066–2075.
   Web of Science ®Google Scholar
• Di Marzio, M, Panzera, A., and Taylor, C.C. (2011), ‘Kernel Density Estimation on the Torus’, Journal of Statistical Planning and Inference, 141, 2156–2173.
   Web of Science ®Google Scholar
• Epanechnikov, V.A. (1969), ‘Nonparametric Estimation of a Multivariate Probability Density’, Theory of Probability and Its Applications, 14, 153–158.
   Web of Science ®Google Scholar
• Fisher, N.I. (1993), Statistical Analysis of Circular Data, Cambridge: Cambridge University Press.
   Google Scholar
• García-Portugués, E. (2013), ‘Exact Risk Improvement of Bandwidth Selectors for Kernel Density Estimation with Directional Data’, Electronic Journal of Statistics, 7, 1655–1685.
   Web of Science ®Google Scholar
• García-Portugués, E. (2020), ‘DirStats: Nonparametric Methods for Directional Data’, R Package Version 0.1.6. http://CRAN.R-project.org/package=DirStats.
   Google Scholar
• García-Portugués, E., Crujeiras, R.M., and González-Manteiga, W. (2013), ‘Kernel Density Estimation for Directional-Linear Data’, Journal of Multivariate Analysis, 121, 152–175.
   Web of Science ®Google Scholar
• Hall, P., Watson, G.S., and Cabrera, J. (1987), ‘Kernel Density Estimation with Spherical Data’, Biometrika, 74, 751–762.
   Web of Science ®Google Scholar
• Hart, J.D. (1985), ‘On the Choice of a Truncation Point in Fourier Series Density Estimation’, Journal of Statistical Computation and Simulation, 21, 95–116.
   Web of Science ®Google Scholar
• Hornik, K., and Grün, B. (2014), ‘movMF: An R Package for Fitting Mixtures of Von Mises-Fisher Distributions’, Journal of Statistical Software, 58(10), 1–31.
   Web of Science ®Google Scholar
• Klemelä, J. (2000), ‘Estimation of Densities and Derivatives of Densities with Directional Data’, Journal of Multivariate Analysis, 73, 18–40.
   Web of Science ®Google Scholar
• Kronmal, R., and Tarter, M. (1968), ‘The Estimation of Probability Densities and Cumulatives by Fourier Series Methods’, Journal of the American Statistical Association, 63, 925–952.
   Web of Science ®Google Scholar
• Laurent, B. (1997), ‘Estimation of Integral Functionals of a Density and Its Derivatives’, Bernoulli, 3, 181–211.
   Web of Science ®Google Scholar
• Lund, U., and Agostinelli, C. (2017), ‘circular: Circular Statistics’, R Package Version 0.4-93. http://CRAN.R-project.org/package=circular.
   Google Scholar
• Mardia, K.V., and Jupp, P.E. (2000), Directional Statistics, Chischester: John Wiley.
   Google Scholar
• Nadaraya, E.A. (1974), ‘On the Integral Mean Square Error of Some Nonparametric Estimates for the Density Function’, Theory of Probability and Its Applications, 19, 133–141.
   Google Scholar
• Oliveira, M., Crujeiras, R.M., and Rodríguez-Casal, A. (2012), ‘A Plug-in Rule for Bandwidth Selection Fn Circular Density Estimation’, Computational Statistics and Data Analysis, 56, 3898–3908.
   Web of Science ®Google Scholar
• Oliveira, M., Crujeiras, R.M., and Rodríguez-Casal, A. (2015), ‘NPCirc: Nonparametric Circular Methods’, R Package Version 2.0.1. http://CRAN.R-project.org/package=NPCirc.
   Google Scholar
• Parzen, E. (1962), ‘On Estimation of a Probability Density Function and Mode’, The Annals of Mathematical Statistics, 33, 1065–1076.
   Google Scholar
• R Development Core Team (2019), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org.
   Google Scholar
• Rosenblatt, M. (1956), ‘Remarks on Some Non-Parametric Estimates of a Density Function’, The Annals of Mathematical Statistics, 27, 832–837.
   Google Scholar
• Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis, New York: Chapman & Hall.
   Google Scholar
• Taylor, C.C. (2008), ‘Automatic Bandwidth Selection for Circular Density Estimation’, Computational Statistics and Data Analysis, 52, 3493–3500.
   Web of Science ®Google Scholar
• Tenreiro, C. (2011), ‘Fourier Series Based Direct Plug-in Bandwidth Selectors for Kernel Density Estimation’, Journal of Nonparametric Statistics, 23, 533–545.
   Web of Science ®Google Scholar
• Tenreiro, C. (2020), ‘Bandwidth Selection for Kernel Density Estimation: A Hermite Series-Based Direct Plug-in Approach’, Journal of Statistical Computation and Simulation, 90, 3433–3453.
   Web of Science ®Google Scholar
• Tsuruta, Y., and Sagae, M. (2017a), ‘Asymptotic Property of Wrapped Cauchy Kernel Density Estimation on the Circle’, Bulletin of Informatics and Cybernetics, 49, 1–10.
   Google Scholar
• Tsuruta, Y., and Sagae, M. (2017b), ‘Higher Order Kernel Density Estimation on the Circle’, Statistics & Probability Letters, 131, 46–50.
   Web of Science ®Google Scholar
• Wand, M.P., and Jones, M.C. (1995), Kernel Smoothing, New York: Chapman & Hall.
   Google Scholar
• Watson, G.S., and Leadbetter, M.R. (1964), ‘Hazard Analysis II’, Sankhyā: The Indian Journal of Statistics, Ser. A, 26, 101–116.
   Google Scholar
• Woodroofe, M. (1970), ‘On Choosing a Delta-Sequence’, The Annals of Mathematical Statistics, 41, 1665–1671.
   Google Scholar
• Wu, T.-J. (1995), ‘Adaptive Root n Estimates of Integrated Squared Density Derivatives’, The Annals of Statistics, 23, 1474–1495.
   Web of Science ®Google Scholar