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Year 2018, Volume: 47 Issue: 2, 273 - 286, 01.04.2018

Abstract


References

  • Babu, G.J., Canty, A.J. and Chaubey, Y.P. (2002). Application of Bernstein polynomials for smooth estimation of a distribution and density function, J. Stat. Plan. Infer., 105, 377–392.
  • Babu, G.J. and Chaubey, Y.P. (2006). Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors, Stat. Probabil. Lett., 76 959–969.
  • Bai, Z.D., Radhakrishna Rao, C. and Zhao, L.C. (1988). Kernel estimators of density function of directional data, J. Multivariate Anal., 27, 24–39.
  • Burgin, M. (2010). Continuity in Discrete Sets. Discussion paper Available at http://arxiv.org/abs/1002.0036v1
  • Butzer, P.L. (1955). Summability of generalized Bernstein polynomials, Duke Math. J., 22, 617–622.
  • Carnicer, J.M. and Peña, J.M. (1993). Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math., 1, 173–196.
  • Carnicero, J.A., Ausín, M.C. and Wiper, M.P. (2013). Non-parametric copulas for circularlinear and circular-circular data: an application to wind directions. Stoch. Env. Res. Risk A. 27, 1991–2002.
  • Eddy, W.F. (1980). Optimum kernel estimators of the mode. Ann. Statist., 8, 870-882.
  • Fernández-Durán, J.J. (2004). Circular distributions based on nonnegative trigonometric sums, Biometrics, 60, 499–503.
  • Fisher, N.I. (1989). Smoothing a sample of circular data, J. Structural Geology, 11, 775–778.
  • Gradshteyn, I.S. and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, 7’th ed., Academic Press: Burlington.
  • Hall, P., Watson, G.S. and Cabrera, J. (1987). Kernel density estimation for spherical data, Biometrika, 74, 751–762.
  • Kakizawa, Y. (2004). Bernstein polynomial probability density estimation, J. Nonparametr. Stat., 16, 709–729.
  • Klemelä, J. (2000). Estimation of Densities and Derivatives of Densities with Directional Data, J. Multivariate Anal., 73, 18–40.
  • Lorentz, G.G. (1997). Bernstein Polynomials, 2nd. ed., AMS Chelsea, New York.
  • Mardia, K.V. and Jupp, P. E. (1999). Directional Statistics, Wiley, Chichester.
  • Mooney A., Helms P.J. and Jolliffe I.T. (2003). Fitting mixtures of von Mises distributions: a case study involving sudden infant death syndrome, Comput. Stat. Data An., 41, 505–513.
  • Petrone, S. (1999a). Random Bernstein polynomials, Scand. J. Stat., 26, 373–393.
  • Petrone, S. (1999b). Bayesian density estimation using random Bernstein polynomials, Can. J. Stat., 27, 105–126.
  • Petrone, S. and Wassermann, L. (2002). Consistency of Bernstein polynomial posteriors, J. R. Stat. Soc. B, 64, 79–100.
  • Phillips, G.M. (2003). Interpolation and Approximation by Polynomials, Springer, New York.
  • Reed, W. J. and Pewsey, A. (2009). Two nested families of skew-symmetric circular distributions, Test, 18, 516–528.
  • Sancetta, A. and Satchell, S. (2004). The Bernstein copula and applications to modeling and approximations of multivariate distributions. Econometric Theory, 20, 535–562
  • Vitale, R.A. (1975). A Bernstein polynomial approach to density estimation. In Statistical Inference and Related Topics, Madan Lal Puri ed., Academic Press, New York, pp. 87–100.
  • Wand, M.P. and Jones, M.C. (1995). Kernel Smoothing. Chapman & Hall, Boca Raton.

Density estimation of circular data with Bernstein polynomials

Year 2018, Volume: 47 Issue: 2, 273 - 286, 01.04.2018

Abstract

This paper introduces a new, non-parametric approach to the modeling of circular data, based on the use of Bernstein polynomial densities. The model generalizes the standard Bernstein polynomial model to account for the specific characteristics of circular data. In particular, it is shown that the trigonometric moments of the proposed circular Bernstein polynomial distribution can all be derived in closed form. Secondly, we introduce an approach to circular Bernstein polynomial density estimation given a sample of data and examine the properties of this estimator. Finally our method is illustrated with a simulation study and a real data example.

References

  • Babu, G.J., Canty, A.J. and Chaubey, Y.P. (2002). Application of Bernstein polynomials for smooth estimation of a distribution and density function, J. Stat. Plan. Infer., 105, 377–392.
  • Babu, G.J. and Chaubey, Y.P. (2006). Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors, Stat. Probabil. Lett., 76 959–969.
  • Bai, Z.D., Radhakrishna Rao, C. and Zhao, L.C. (1988). Kernel estimators of density function of directional data, J. Multivariate Anal., 27, 24–39.
  • Burgin, M. (2010). Continuity in Discrete Sets. Discussion paper Available at http://arxiv.org/abs/1002.0036v1
  • Butzer, P.L. (1955). Summability of generalized Bernstein polynomials, Duke Math. J., 22, 617–622.
  • Carnicer, J.M. and Peña, J.M. (1993). Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math., 1, 173–196.
  • Carnicero, J.A., Ausín, M.C. and Wiper, M.P. (2013). Non-parametric copulas for circularlinear and circular-circular data: an application to wind directions. Stoch. Env. Res. Risk A. 27, 1991–2002.
  • Eddy, W.F. (1980). Optimum kernel estimators of the mode. Ann. Statist., 8, 870-882.
  • Fernández-Durán, J.J. (2004). Circular distributions based on nonnegative trigonometric sums, Biometrics, 60, 499–503.
  • Fisher, N.I. (1989). Smoothing a sample of circular data, J. Structural Geology, 11, 775–778.
  • Gradshteyn, I.S. and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, 7’th ed., Academic Press: Burlington.
  • Hall, P., Watson, G.S. and Cabrera, J. (1987). Kernel density estimation for spherical data, Biometrika, 74, 751–762.
  • Kakizawa, Y. (2004). Bernstein polynomial probability density estimation, J. Nonparametr. Stat., 16, 709–729.
  • Klemelä, J. (2000). Estimation of Densities and Derivatives of Densities with Directional Data, J. Multivariate Anal., 73, 18–40.
  • Lorentz, G.G. (1997). Bernstein Polynomials, 2nd. ed., AMS Chelsea, New York.
  • Mardia, K.V. and Jupp, P. E. (1999). Directional Statistics, Wiley, Chichester.
  • Mooney A., Helms P.J. and Jolliffe I.T. (2003). Fitting mixtures of von Mises distributions: a case study involving sudden infant death syndrome, Comput. Stat. Data An., 41, 505–513.
  • Petrone, S. (1999a). Random Bernstein polynomials, Scand. J. Stat., 26, 373–393.
  • Petrone, S. (1999b). Bayesian density estimation using random Bernstein polynomials, Can. J. Stat., 27, 105–126.
  • Petrone, S. and Wassermann, L. (2002). Consistency of Bernstein polynomial posteriors, J. R. Stat. Soc. B, 64, 79–100.
  • Phillips, G.M. (2003). Interpolation and Approximation by Polynomials, Springer, New York.
  • Reed, W. J. and Pewsey, A. (2009). Two nested families of skew-symmetric circular distributions, Test, 18, 516–528.
  • Sancetta, A. and Satchell, S. (2004). The Bernstein copula and applications to modeling and approximations of multivariate distributions. Econometric Theory, 20, 535–562
  • Vitale, R.A. (1975). A Bernstein polynomial approach to density estimation. In Statistical Inference and Related Topics, Madan Lal Puri ed., Academic Press, New York, pp. 87–100.
  • Wand, M.P. and Jones, M.C. (1995). Kernel Smoothing. Chapman & Hall, Boca Raton.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

J.a. Carnicero This is me

M.p. Wiper This is me

M.c. Ausín This is me

Publication Date April 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 2

Cite

APA Carnicero, J., Wiper, M., & Ausín, M. (2018). Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics, 47(2), 273-286.
AMA Carnicero J, Wiper M, Ausín M. Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics. April 2018;47(2):273-286.
Chicago Carnicero, J.a., M.p. Wiper, and M.c. Ausín. “Density Estimation of Circular Data With Bernstein Polynomials”. Hacettepe Journal of Mathematics and Statistics 47, no. 2 (April 2018): 273-86.
EndNote Carnicero J, Wiper M, Ausín M (April 1, 2018) Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics 47 2 273–286.
IEEE J. Carnicero, M. Wiper, and M. Ausín, “Density estimation of circular data with Bernstein polynomials”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 2, pp. 273–286, 2018.
ISNAD Carnicero, J.a. et al. “Density Estimation of Circular Data With Bernstein Polynomials”. Hacettepe Journal of Mathematics and Statistics 47/2 (April 2018), 273-286.
JAMA Carnicero J, Wiper M, Ausín M. Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics. 2018;47:273–286.
MLA Carnicero, J.a. et al. “Density Estimation of Circular Data With Bernstein Polynomials”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 2, 2018, pp. 273-86.
Vancouver Carnicero J, Wiper M, Ausín M. Density estimation of circular data with Bernstein polynomials. Hacettepe Journal of Mathematics and Statistics. 2018;47(2):273-86.